Qfi 467 
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Glass 



n u A f H 



Book 



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mm'n 



comcmxyt^i^ 



HE CIRCLE 



THE ELLIP 



A PiSOUSSiON OF THE PROPERTifcS 



OF' TKS- 



<ftrmg' 




ittje mA iU 




WITH 



A CRITICAL EXAMINATION OF 

BRAIO ANALYSIS &o. 



" If a better s^iUm's thine, 

Impart ifrfeakly, or make us® cf mins 1' 



BY LAWRENCE 8, iENS 



5i:iGKa'S SlEii-M EGOS' AIv'D JOB FKIIiTI?rS E.3TA£U3ffikfa;:fT-. 




yrs^. 




I> 



/ 



s^ 



1 



SOME IIVFORMATIOW 



GOIMGERNING THE 




1^ 



bl u 




# 




u 



Omission. — In the Dedication of this work, the name 
of William Roberts, of BurkE County, Ga., was omitted 
by mistake, from the list of the Graduated Class of 185B 
of the University of Georgia. 



Ill 



TO THE PUBLIC. 



In the consideration, that, a portion of the matter contained ia 
this Pamphlet l>€ing, a,bout a year or more ago, printed and distri- 
buted by me, among my friends and others, in the principal Literary 
Institutions of the country, North and South, and elsewhere, I fm\e 
thought it best upon this occf sion^ hj way of an introduction^ to 
make a few remarks. The subject, here presented in a more complete 
and satisfactory manner, is one of va^t importance and great celebri- 
ty ; one upon which the Time, Ingenuity and Learning of near and 
remote ages have been expended in vain ; one upon which the ad- 
vancement and success of various modern sciences are solely depen- 
dent ; and one with which ths factSj observations 8.nd principles 

of the great science of Astronomy are, if not entirely, intimately 
connected. It is a subject which has long puzsled the clearest in- 
tellects of the world ; and one which has perhaps received more re- 
flection and attention than any other ever offered to scientific in- 
vestigation. So much has the belief ga'ned ground of the impossi 
bility of its solution, that to be known as '' one trying to square the 
Circhj'^ becomes at once a subject of jest and ridicule, to many of 
those who iave never opened a book on Geometry, to some of 
those who have read, but never un5erstoodv the rudiments of 
the science — to few of those who have tolerable ideas of the Prob- 
lem, and to none of those who know the Piinciples of Geometry. 

In offering this System of Principle and RuUs to the Public^ I place 
myself undoubtedly in a peculiar situation, not only in attacking 
the Principles aad Rules supported by the highest Geniuses of the 
Ancient and Modern worlds, but also in preseutiog a nev> system to 






the criticism and prejudice of the Public. To give ,,r«,«/A f „ 
■position and point to my argument r h. . ' ""^ 

y argament, I here make the following offer: 

roa,,,,erson.ko,etecf.anyErrorin t,e System of Pnnc^ples and 

.^ro^.«,M«.«M,^,„„rfe.«..-./^., of the Principle vpon v>Meh the 
.a^a argument is conducted, J U,,a my.elf to „ the ^m of 
One Thousand Dollars. - 

Signed Lawrence S. Benson. 

and r ?" rr' " '"° ""'^' ' P'"^'''^^ ^° ^'^^^^^ '^^ *'-« 
and attention of the Learned ; surely Th..h is a G..kdon well 

worthy them. ' 

Mezula, S. C, March, 1862. 



???»*X^i^ 




SCIENTIFIC DISQUISITIONS 



00KC3&E?^1?^Q 



T 










AND 



TIIJi: ELLIPSE 



c- 



A DISCUSSION OF THE PROPERTIES 



OF THE 



^ttaigbt ^\m m& iU &m. 



WITH 



A cnrncAh examination of the algebraic 

ANALYSIS. 



o 



' If a. better syste/jii^ t!iine. 
Impart ir traEkiy. or maka use of mine/ 



-o 



BY LAWKENCE S. BENSON 



?%' 



AIKEN, S. C. 



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X>EIXDIO^^TIO 3Sr. 



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52 



I DEDICATE TO THE 



fliBUlfSi CiilS« 



ill 



or THE 









Robert C. Humbek, 
JoiiN R. North, 
William H. Priciiett. 



Joseph Taylor. 



Gilmerb R. Baxks, 
Augustus P. Hodo, 
William T. Mitchell, 
James M. Oliver, 
G. Gilmere Raglan, 



As a leuiemb.rancer of tlios,e pleasant moments, we passed so of- 
ten together, and of those delightful associations of '• Old Frank- 
lin," which cluster around the heart even amidst the Sthfes upon 
the world's arena. 



Entjere(J according to:^ct of C^ngrt;?8 of tlie €or^ederate States 
of America, on the sixth of March, 16(>2, b}^ 

LAWRErsCE §. r>ENSON, 

in the Clerli'o Office, of the Confederate Court, of South Carolina, 
ill Charleston. 



w 



r«i 



o 



PREFACE. 



lor the approval, of an inteiligeui Public, it may be necessary to 
preface l])5it. conscious of t lie riftk,. and ieclingliie reiipoUvSibiUiy, oi 
I may say. the audacity of .*:jcn an indertaking, I have, relying ou 
the iiUegiily of my motives, and the conviction oi right, thrown my- 
self upon th.e waves of bia?^ and prejudice ; and hope that my stru?- 
•rles may vix'm formeth.e timooth and gentle curvents of rejection and 
iiidgmeni. Nothing o;reat# has e<'er been won by faintheartedness . 
we mnstlanch our barque upon the lareaiu, in spite of the ws^ve^. ih^ 
winds and ihe teirspesi> whch may l>e gathering arouud. These re- 
mark:^ have been sugiie>ied by conlenp'aring the purport of these 
Disquisitions. 1 li a ve attacked tiie system fojuided by the greatest 
ancient n«athe?naiician. Archiaiedfts : sv.ppor'.ed by the iiln«trioiis 
Newton, Leibnitz. Lu^^ace. Huygens and others : and lioary wrtk 
ihe silver hair.« of more than twotlioitsadd yei^rs. 

It is not without great .svi^icitiide that 1 oiTer the.-;e pages at the 
Tribunal of .Science- But I feel af:f?nred. that decitious" coming from 
that Bejich are ct^tabli.-hed upon Truth : and wlicn once eyta.Mi^lied 
iiiere. are as tiirn.and unalverabie as the Universe itself 

The trutli of a principle is never so well established as whf*n it i^ 
»'ousi»tent iu every application that it may be necessary to use it. 
Every Science, whether Political, Moral. Physical, Metaphysical, ot 
-Matheniatica!, reqnireis in its foundation some elementary property, 
givmg a« the corner stone of an edifice, a particular character and 
use te the supen^tructure that may be raised upon it. This eieiieu- 
tary property is (lie principle of the science, which, by iis consistence, 
applicability, and ns^, determine it*» trutli. and establish its fundimout- 
aliiy- 

Science is the kev bv which we unlock t])e mysteri-e^-of Nature 
and bfhoid the vast Uuiverse ic, ail its nakf d realities. It is more ; i4 
IS the great light by which we can penetrate the darkness of the far 
distant past : under stand rhe pre:>ent ; and fathom the eternity of 
the fulure. f''ience teaches us the wisdom and power v^ God. 
There is as much syj^temritic skill displayed in the creation of ari 
auimaloufe, as there is in the coDstruCtiou of a planet. The vital 
Jty of the one is as grand as the existence of the other. The pow- 
^r which condenses the Unid, is the same ai^ that which gives foiife 
and shape to the uiiiverse. Science is the embodiment of wi^doiM 
and power. God is the great fountain-head of science. 

Not wi.-^lung to anticipate any of the points discussed in the 
lollowing pages, I will si nply snlmiit thep*,^ Dis^qnisitiouR ra the 



earnest coiibideiation oi" those who are intorcsted in these snhif^rt^ - 
with the hope, that my eflbrts may meet with that receDtion due 
to such important subjects ; and with the desire, that if w^ono- 
condemn me, and if-right, uphold me. "^-^"cj 

In my researct^s, I am much indebted to ^' Brande'sEncvclo 
pasdia ; though much at variance with it, T appreciate the industrv 
and zeal exhibited m that valuable compendium of science. 



Mbzula, Nov. 1861. 



LAWRENCE S. BENSON. 



Hi 



THE PITH. 



• •■♦- 



I HOLD tkat the solution published by me in this pamphlet, 
doth contain the correct version of that celebrated problem enti- 
tled the Quadrature of the Circle: for the reason that the only objec- 
tion to my version of the said problem, is where I show that the 
square equal in area to the circle, is the intermediate square be- 
tween the circumscribed and inscribed squares of the circle. I 
hold that I am just as much entitled to the use, power and influ- 
ence of evidence in my version of the said problem, as other Ge- 
ometers are to evidence in their demonstrations of any geometrical 
problem whatever : since all their demonstrations, and the science 
of Geometrv, are based upon evidence— and it would be inconsist- 
ent with the prvieipks of the science, to deny me the virtue of 
evidence. 

I hold that my argument for the solution of the said problem is 
Irrefutable ! and as such is entitled to all the credit, virtue and 
value of any correct and well substantiated argument. I hold 
furthermore, that my argument, being established upon evidence, 
and irrefutable, is as binding upon scientific investigations as the 
axioms of Euclid, the laws of Kepler or the hypothesis of New- 
ton. And finally, I hold that my solution is more geometrical, 
and consequently, more scientitical, than any solution of the said 
problem before ofiered to the investigation of the scientific world, 
because every solution heretofore offered has been tinctured either 
with algebraic formula or arithmetical quantities, or has confoun- 
ded the properties of the straight line with those of the curve. 



m>amt^mmmma^9nmmi>K)m^m 



SOME INFORMATION 



CONCERNING THE CIRCLE. 



I2f PfiOFOSlTION XI, Book V, DaVIES^ LsGENDREf- 

where the surface of a regular circumscribed, and of a 
similar inscribed polygon are given, we can find the sur- 
faces of regular circtim scribed and inscribed polygons 
having double the number of sides. And as we double the 
number of sides of these regular and similar polygons, we 
can still be able to find the surfaces of the new polygons. 
We perceive, as we double the number of sides of each 
new circumscribed and inscribed polygon, that the differ- 
ence of their respective surfaces diminishes; and if we 
continue the calculations oaI injinitum, the diiference will 
decrease to an indefinite quantity. At the same time, the 
difference will never be destroyed : but to all practical 




defined to be a ficrure hounded by a curve line. A curve 
line has been defined to be a line which chan2:es its direc- 
tion at every point. A straight line is one that maintains- 
the same direction between any two of its points. Now 
a polygon is a figure boufided by straight linesy— each line 
constitutes a side to that polygon. A polygon consists of 
three or more sides — never less than three sides. A cir- 
cle is not a polygon — being bounded by one curve line, it 
has less than three sides. Now, in Proposition XIL same 
book, the circumference of the circle ma}^ be conceived to 
be the thousandth part of a hair in thickness, still it will 
be the circumference of the circle ; and the perimeters of 
the polygon may be conceived to be of the same thickness. 
The number of tlie sides of the polygons may be doubled 



•SOME INFORMATION' 



a^ infinitwn^ and the surfaces of the polygons may be com- 
puted to the millionth place of decimals, so that the differ- 
ence of the surfaces will be absolutely nothing— still the 
perimeters of the polygons will -never coincide with the 
circumference. For the side of a polygon can never be 
reduced so small as to lose its character of a straight line. 
A straight line and a curve line can never coincide. It is 
preposterous to suppose a curve line to be made of a bro- 
ken straight line. 

According to Proposition VIII, same book, the area 
of a polygon is equal to its "perimeter multiplied by one 
hal^the apothem of the polygon. Now, this is true, as 
regards the polygon. Scholium 2, Proposition XII, Book 
Y^ supposes that the area of the circle is equal to its cir- 
cumference multiplied by one half its radius. But the 
circle is not a polygon ; and what is true as regards a 
polygon is not so as resjards the circle. The same Scho- 
lium further adds, that a circle is a regular polygon of an 
infinity number of sides. Now, a circle is a finite figure, 
consequently its side or sides must be finite. An hyper- 
bola is an infinite figure. Therefore, Proposition XV, 
Book V, is not tenable. 

In Proposition XVI, Book V, there is an incongrui- 
ty of terms. It begins by premising that the area of a 
circle is equal to the product of the square of radius by^ 
the ratio of the diameter to the circu^fererce. It ends 
by asserting that the area of a circle whose diameter is 
two, is equal to the circumference of a circle whose diam- 
eter is one — or let the Proposition speak for itself — we 
have this Proposition : I : Pi ; : 2 C A : cir. C A : or, 
diam. : cir. : : diam. : cir. ; Now, the value of diameter 
of second pari of the Proposition, is given in the Scho- 
lium—which is 2 — therefore, 1 : Pi:: 2 : cir. C A : or 
cir. C A=2 Pi\ or circumference of circle whose diam- 
eter is two^ is equal to twice the circumference of a circle 
whose diameter is one. Now, this knowledge may, in 
some cases, be valuable, but in this it rather impedes than 
assists the demonstration of the proposition. And again : 

Area C A -= Pi X G~K' Bv Scholium : C A =- I ,\ 
Area A = Pi : or area of circle whose radius is unity, 
is equal to circumference oi Q\vi^\Q whose diameter is unity. 



1^ 
It 



COJStEPcNING THE CIRCLE. 9 



We have again : Pi = 3.141592G, now 3.1415926 -j- is 
,, the area of a polygon whose apothem is unity — in other 
J wordsj the circuYHJerence of a circle whose diameter is uni- 
,f ty, is equal to the area of a polygon whose apothem is 
; unity. ^This is plainly an assumption based upon an as- 
sertion. 

In reo-ard to Propositions XV. and XVI., Book Y., it 
may be argued, the correctness of the premises, and the 
plausibility of the demonstrations, from the equality or 
similarity of the conclusions. If we examine these pre- 
mises, we will soon see why thej coincide. Proposition 
XV. premises that the area of the circle is equal to the 
product of half the radius by the circumference : now the 
circumference is asserted to be the product of the diame- 
ter by the number 3.1416. The diameter is twice radius. 
Therefore, area of circle is equal to the product of 3.1416 
by twice radius, by one half radius. Proposition XVL 
premises that the area of the circle is equal to the pro- 
duct of 3. 1416 by squ.^re of radius. The square of a 
number is equal to twice the number multiplied by one 
half the number. Here we have very plainly the number 
3.1416, multiplied in each case, by the same quantity. \% 
it anythino- extraordinary, that the product should be the 
same? Now, Proposition XY. is wrong; therefore, its 
equal, Proposition XYI., is wrong also. '' Things equal 
to the same thing, are equal to one another.^' 

It has been argued, to support the premise of Propo- 
sition XYI., that the number 3.1416 is the ratio of the 
diameter to the circumference, from the fact that th^ radius 
when applied to the circumference, vyill subtena an arc 
equal to one-sixth of the circumference; therefore, it is 
concluded that the circumference is three times the diam- 
eter, i^ow, the mean proportional of the diameter and 
radius subtends an arc equal to one-fourth the circumfer- 
ence ; and by the same principle, the circumference is four 
times the mean proportional. The diameter subtends an 
arc equal to one-half the circumference ; therefore, by the 
, same principle, the circumference is twice the diameter; 
wbich does not agree with above. 

Proposition XYI., Book Y., claims that Fi = 3.1- 
4169-26 ; and Fi by its own representation, is the circum- 



10 



^U.\Ji-: lNF0il3JAT;O.\ 



"nir,v, IS equal to o. Ul -)9v") X -^ "'ueror is 

,.te he... Even gnuui.g ,hf, 'i^^bJined- bat "t 
.ouputing tbc area o ' d cle Vo Tf - "' ^''''^'''' f 

ivDOWD, anil that irv;.hn..^^ g ^-^ suthtieLt that they are 

-..auv predoHs .non/o'r:;! h^v^ hi ^.^r.S^i'^t" ^-' 
on it5 ramparts, with error insr-Wbed 1";^ 'l '"?= 

exteouation, I will idd tl 7 * n ■ ' ^^^^f^r. Jn 

ic...» ..• .„/.: :^;r:r,'. i7.s?:r 'r:,r«.:t 



CONCERNING THE CIRCLE. 1 I 



or}' niidertook to demoDstrate that the definite quadrature 
is impossible ; Huygens, however, objected to his reasoning; 
and the opinion of Geometers seems to be, that no satis- 
factory demonstration has yet been given,- of the absolute 
impossibility of solving the problem. It has been proved 
by Lambert and Legendre, that the ratio of the circumfer- 
ence to the diameter and its square, are irrational num- 
bers. But ^'rational quantities may be susceptible of 
geoni^tricrd construction, ; and if a straight line^ equal to 
tlie circumference of a circle^ having a given radius, 
could be constructed geo'inetrically^ the quadrature of the 
circle would be accomplished^ although the length Of the 
line could not be expressed by a finite number. With re- 
gard to the indefinite quadrature, Newton has demonstrat- 
ed in his Principia, (thou2:h in a manner not altogether 
unobjectionable,) that no curve which returns to itself, like 
the circle and ellipse, is susceptible of it." 

*^ Numerous pretenders to the discovery of the quad- 
rature of the circle, have appeared at various times, and 
they occasionally present themselves even at the present 
day. They are only to be found among those who have 
an imperfect knowledge of the principles of Geometry ; and 
when their reasonii^g happens to be intelligible, their pa- 
ralogisms are, in general, easily detected. With a view to^ 
discourage the futile attempts, so frequent!}^ made on this 
and similar subjects, the Academy of Sciences of Paris, in 
1775, publicly announced, that it would not examine, in 
future, any paper pretending to the quadrature of the cir- 
cle, the duplication of the cube, the tri section of an angle, 
or the discovery of the perpetual motion ; and shortly after 
the Royal Society adopted a similar resolution." See 
Quadrojture of the Giicle. 

From the above, it will be perceived.that the opinion 
of Geometers, is not of the impossibility of the quadrature 
of the circle — and that, though the. Academy of Sciences 
and the Royal Society passed resolutions as above, it was 
. not with the opinion of the impossibility of the problem : 
but for the purpose of self defence. Now, it is seen, that 
' • if a straight line equal to the circumference of a circle 
having a given radius^ could be constructed geometrxally^ 



12 SOME Information 



the quadrature of the circle -would be accomplished^ aU 
though the length of the line coy Id not be exjyressed by a 
finite mimberS'^ To the diagram annexed (No. I,) let us 
turn our attention. 

Granting A B C D, to be a perfect circumscribed 
Si^uare, and 1 P K L, to be a perfect inscribed square, 
having its sides parallel to the square A B C D; and, 
supposing tlio circle to be flexible material — wire, for in- 
stance — '.f e would find that in straightening the curvature, 
thai the circumference will form the square E F G H ; 
tor the reason, that the points I and *P, -will extend up* 
wards towards & and F, aud th^poinl M, downwards to* 
wards Ft, and that the point M, will fall just so much a& 
the points i and P will rise — being one-half the difference 
between the squares A B G D, and I P K L. For it 
is evident, that the are, when straightened, will be equal 
to the line E F, from the fact, that the arc varies from the 
chord, the distance M R ; aud it is very palpable that the 
arc, when straightened, will extend the distance M R, be- 
yond the extremity of the chord I P , and the line E F 
is equal to the chord I P, plus the distance M R. This 
principle is deduced from the relations between the squares- 
and the circle. Ft may seem reasonable and look plausi- 
ble, that this principle should hold w-ien. applied to any 
arc less thau a quadrant; but there it becomes very plain, 
that when this principle is applied to any are less than a 
quadrant, this'principle is then a different principle: be- 
cause this principle is founded upon the intimcUe and 7iear 
relations between the squares and the circle — the other, 
between the polygons and the circle. Now, the relation& 
in the one case, are different from the relations in the 
other. The relations between the squares and the circle 
are these : the circumscribed square is the square of the 
D J i^ meter; the inscribed square is the square of the mean 
proportional of the diameter and radius. The altitude 
of the segment contained by the quadrant and its chord, 
is one-half difference between the diameter and the mean 
proportioual of the diameter and radius. These are in- 
timate relations between the squares and the circle, which 
dc not exist between any regular and similar polygons oi 



CONCERNINQ THE CIRCLE* 



13 




Diagram No I 



14 



SOME INFORMATION 




Diagram SHo^ 2. 



CONCERNING THE CIRCLE. 15 



more or less than four sides aud the circle. Principles 
deduced from different relations, are evidently different. 

Now the lines E F, F H, H G, and G K, are equal 
to the circumference ; and therefore/ their sum is the line 
so much desired, and consequently, the area of the square 
E F H G, is equal to the area of the circle. 

In Diagram JS^o. 2, let it be granted that the circle is 
equal to the one in Diagram No. I, then we will have as 
follows: [See Diagram No, 2.] 

Having been shown in Diagram No. I, that the square 
E F H G, is precisely between the circumscribed ^nd 
the inscribed squares ; it follows, that its equal, the circle,, 
must also be between the circumscribed and the inscribed 
squares — or, in other words, the circle divides the distance 
between the two squares also ; consequently, the segments 
A B/ &c., are equal to the figures x\ B E, &c., or 
the segment A. B, is equal to one-half the triangle 
ABE, and the triangle A 13 E is one -half square of ra- 
dius ; thereiore, the segment A B, is one-fourth square 
of radias Now, the area of the circle is equal to 
the inscribed square A B C D, plus the segments. And 
A B D, is equal to twice square of radius ; because it ' 
contains twice the number of equal triangles. The four 
segments, being each equal to one fourth of square of ra- 
dius, are equal to square of radius ; consequently, the area 
of the circle is equal to three times square ef radius. 

To construct tbe geometrical square, equal in area to the 
circle, we will proceed as follows : in the Diagram No. 2, 
we have the squares E F G H, and ABC D. Kow^ 
E F G H, is double A B C D — having double the 
number of equal triangles; and A B CD, for like rea- 
son, double A M C F, the square of radius .'. we get this 
proportion : EFGH:ABCD::ABCD:AMCF, 
or squaro of diameter is to A B C D, as A B C D is to 
square of radius ; therefore A B C D is the mean propor- 
tional of the square of diameter and the square of radius^. 
By Diagram No. 1, the square equal in area to the circle, 
is just half the difference between the squares, circum- 
scribed and inscribed — but the inscribed square is the 
mean proportional of the square of diameter and the square 



/ 



16 So.viE iNFOr.AiATlON 



of radius. Therefore, a line ev|ual to the mean propor- 
tional of the diameter and radius, plus one-half difference 
of the diameter and the mean proportional, will be the 
length of the side of the square, equal in area to the circle. 

Scholium 1. The chord of the quadrant of the circle is 
the mean proportioiaal of the diameter and radius. 

Scholium' 2. The segment contained by the quadrant 
and its chord, is equal to one- fourth of the square of the 
radius. 

The area of a sector is contained within two radii and 
an arc of the circle- Now, since the arc is a part of the 
circumference, and bears a proportional part oi it accord- 
ing to its length — and the sector is a part of the area of 
the circle, and is governed by the arc — we can adopt the 
arc as the means of measure in calculating the area of the 
sector. For instance, if we suppose the arc to be one de- 
gree, which being the three hundred and sixtieth part of 
the circumference, the area of the sector will be the three 
hundred and sixtieth part of the area of the circle ; or 
again, if the arc is ninety degrees, the area of the sector 
will be one-fourth of the area of the circle, and so on. 
Now wh«n the radius of any circle is known, the area can 
be obtained, and that of any sector in that circle, by sim,- 
ply computing the number of degrees in that arc. If the 
arc contains degrees, minutes and seconds, t^e minutes can 
be calculated to the paits of a degree, and the seconds to 
the parts of a minute. 

To find the area of any segment, we would simply pro- 
ceed to find the area of a sector haviuir an arc eanal to the 
arc of the segment ; then subtract from this sector the tri- 
angle contained by the radii and the chord of the segment 
— the remainder will be the segment. Now, in this tri- 
angle, two sides and the included angle will be known, the 
triangle being isosceles^ the other angles can readily be 
found, which will leave, then, only cue side of the trian- 
gle unknown. 

The rule for finding the length of any arc which could 
be assigned, can be illustrated as follows : —The arc I M P 
in Diagram No 1. has been shown to be equal to I P -j- 
P Z = I Z ; but P Z = R M, the altitude of the segment 
IMP; consequently, the length of the quadrant is equal 
to the base of the segment plus the altitude. Since this 



y 



CONCEPcNING THE CIRCLE. 17 



rule for finding the length of an are, is dependent on the 
quadrant of the circle, this rule must always be referred 
to the quadrant ; thus, the quadrant contains a fixed and 
known number of degrees, and each degree is a proportion- 
al part of the quadrant. When the arc is less than the 
quadrant, as many degrees as it is less, the arc will be so 
many proportional parts less —thus : if the arc is 45 de- 
grees, it will be tf parts less than the quadrant ; or, if it is 
75 degrees, it will be W parts less, and so on. If the arc 
is more than a quadrant, it will be *so many parts more— 
thus : if the arc is 120 degrees, it will be t^ parts more 
than the quadrant ; if 190 degrees, it will be ttvice Kud 
"H parts more, and so on. This rule is applicable when 
the ratio between the circumference and diameter is known, 
because then the circumference can be found ; and the 
quadrant will be one- fourth, each degree one-three hundred 
and sixtieth part of the circumference ; or, one-ninetieth 
part of the quadrant. Upon the same principle, the length 
€'f any arc can be obtained geometrically. To find the ra- 
tio between the circumference and diameter, we will pro- 
ceed as follows : we will form the circumference into a 
square — we can obtain the area of the square by the rule 
for the circle — thus : if the radius be four inches, the area 
of the circle will be forty-eight inches — then the square 
root of forty- eight will be the length of the side of the 

square: 4 V 48 will be the perimeter of the square, equal 
to the circumference of the circle — then the ratio will be 
3.4641 --|-. As an evidence of the correctness of this^ 
rule, if we examine the formation of the curvature of a 
quadrant, we perceive that the quadrant varies from the 
chord tho distance of the altitude of the segment. Now, 
it is very palpable that the arc, when straightened, must 
extend the distance of the altitude beyond the extremity 
of the chord. 

To construct a triangle equivalent to any sector, we 
would proceed thus : take, for instance, a sector equal to 
one-fourth of the circle— now, a triangle equal to that sec- 
tor, must be equal to one-fourth of the square E F- G H, 
in Diagram No, I. This triangle is formed ia the follow- 
ing iBanner : The base is formed by adding to the base of 
the triangle contained by the radii and chord of the sector, 



/ 



•8 SOME INFORMATION 



_ i_ 



the altitude of the segment of the sector ; and the altitude 
is formed by adding to the altitude of the triangle con- 
tained by the radii and chord, one-half of the altitude of 
the segment of the sector. According to Brande's Ency- 
clopaedia, the sector is equal to the triangle formed by the 
radius as an altitude, and a line equal to the arc as a base. 
This is upon the erroneous supposition already exposed, 
viz : that a circle is a regular polygon. If the arc of the 
sector is greater than. a quadrant, apply the rule —the sum 
of the lengths will be the base of the triangle : the altitude 
will be common ; therefore, that can easily be obtained. 

If we apply the rule for finding the length of an arc, 
to the arc M P, in Diagram No. I, we will find it equal to 
the line F*. Now, if the point M falls more or less than 
the points I and P rise, the arc M P cannot be equal to 
the line F ; because, in the first place, the principle from 
which this rule is deduced would be wrong ; and in the 
second place, its fallacy would be perceptible by illus- 
tration. * 

If the area of any magnitude, bounded by regular 
curved lines, wish to be known — we will take the case 
where the sides convex — we will construct a rectilinear fio:- 
ure, by connecting the extremities of each arc by straight 
lines, then we will have the area of the magnitude equal to 
the area of the rectilinear figure plus the segments. The 
rule for finding the area of a segment, has been given ; 
therefore, add the sum of the areas of the segments to the 
area of the rectilinear figure, and we will have the area of 
the magnitude. When the sides of the maornitude con- 
cave, we will proceed as above, but subtract the areas of 
the seojments tVom the rectilinear fiaure formed. 

When we have a magnitude with some of its sides 
concaving and some convexiog : we will form the rectilin- 
ear figure as described above; and add the sum of the 
areas of the segments which convex, and subtract the sum 
of the . areas of the segments which concave ; the result 
will be the area of the marrnitude. 

\\i pursuing successfully' the above experiments, we 
must have certain things known — or, in other words, the 
foundation upon which to build the superstructure. We 
do not ask too much; the carpenter, to bn^'ld his house, 

* F = M E F. 



it; 



CONCER-XING THE CIRCLE. 



19 



must have his complement of tools — the same with the 
mason, smith, or any other department of trade or profes- 
sion. We ask for our tooh to carry on our operations. 

DiNosTRATus has the credit of effectinn; the rectifica- 
tion and quadrature of curvilinear spaces mechanically, 
bj means of a transceudal curve, which is termed the 
quadratrix. (See Diagram in Brande's Encjclopfedia, un- 
der '^ Quadratrix. '') In applying this curve to effect the 
rectification and quadrature of the circle : there will be a 
difficulty to find the precise point where the quadratrix 
A P Q meets the radius G B: to determine Q ; and 
prove Q to be the point. By mechanical contrivance, Q 
may be the point ; but it could not be substantiated by de- 
monstration. Science is theoretical, and art is practical, 
The contrivance may suit the art, but to the science it 
would be useless. If we could know the exact point by 
demonstration, we could find the proportion between C Q, 
the radiu? C B, and the quadrant A B. C Q is supposed 
to be the third proportional of the radius and the quadrant. 
Hence the proportion AB:BC ::BC:CQ.-. AB = 

CQ . Now, if the point Q could be designated accurately, 

C Q — being; a straio-ht line — could be determined : and the 
'-^ ^ . , ,-t 

rectifi-cation of the arc A B weuld be arithiiiettcauy ac- 

complished. The equation C A B = .7^ is obtained by 
supposing the area of the quadrant to be equal to the arc 
multiplied by one-half radius — which ^ wrong in toto. 

The quadratrix of Tschimhausen .amounts to the same 
as above. We have no doubt that the quadratrix could 
satisfy the problem of the rectification of the curve, if the 
mechanism could be perfected so as to give the exact point 
Q — C Q beinn: a straight line, could be determined ; and 
the proportion between B C, G Q, and A B might be ob- 
tained. But knowing* the leno-th of A B, would not en- 
able us to find the quadrature of the circle. Because, we 
could not by means of the arc and radius, determine tte 
area of the circle; as we could the area of a regular poly- 
gon, by knowing the apothera and a side. 



20 SOME INFORMATION 



THE QUADRATURE OF THE ELLIPSE. 



Tub nature of an ellipse is such, that if two straight lines 
l)e drawn to any point in its periphery, from two fixed 
points in its transverse axis, the sum of these two straight 
lines, will be always equal to tho transrerse axis. (See 
Davies' Analytical Geometry, Book JY.) Now, if the 
two lines be drawn to the extremity of the conjugate axis; 
it is very evident, that each line will be equal to the other; 
and, conse(|uently, equal to one half the transverse axis. 
If the square be completed by drawing similar^ lines to the 
other extremity of the conjugate axis, the conjugate axis 
will be the diagonal of tbat square. By Diagram No. 2, 
it has been shown that the line B A, is the mean propor- 
tional of E F and E A. And E A is one-half E F, and 
equal to the other side of triangle E A B. But the tri- 
angle E A B, is a right angled. Therefore, when the two 
sides of a right-angled triangle are equal, the hypotenuse is 
the mean proportional of the sum of its sides and one of 
its sides ; consequently, the conjugate axis is the mean pro- 
portional of the transverse axis, and one half the transverse 
axis. And will be the side of the inscribed square of the 
circle, of which the transverse axis is the diameter. 

When the extremities of the transverse and coniuirate 
axes are joined by straight lines, t'le rhombus is formed, 
the square of whose ^de is equal to the sum of the squares 
of one-half the transverse axis, and one-half the conuio:ate 
axis. (Proposition XL, Book IV., Da vies' Lcgendre ) The 
perimeter of the rhombus is'equal to the perimeter of the 
square formed by its sides. The square is the inscribed 
square of the circle, of which the side is the mean pro- 
portional of the diameter and radius. Having this side^ 
we can make it the hypotenuse of aright-angled isosceles 
triangle: then, the vertex of this triangle will be the cen- 
tre, and either side of the triangle will be the radiua of 
the circle. The hypotenuse will.be the chord of the quad- 
rant, or the mean proportional of the diameter and radius 
of ^e circle So, having the mean-proportionai, we caa 
find the diameter and radius, (of which, the square of the 



COXCEKiSING THE CIRCLE. "21 



mean-proportional is the ioseribed square; whose perime- 
ter is equal to the perimeter of the rhombus,) in the fol- 
lowing manner: the square of diameter being four times 
the square of radius ; the square of mean proportional, 
multiplied by two, will produce the square of diameter ; 
and the square of mean-proportional, divided by two, will 
produce the square of radius — thus: suppose 15' is the 
square of mean-proportional, or square of side of the 

rhombus—then, 2(15) : 15 :: 35 : ^^ .*. 2(15) is the 

square of the diameter, and — o~ is the square of radius. 

An ellipse is also formed by a plane intersecting the 
cone obliquely to the sides and base ; and to the plane of 
a sub-contrary section, The mechanical formation of the 
ellipse, is essentially the same in principle as the forma- 
tion of the circle: but, owinoj to the circumstances of their 
formation, this principle is differently applied. The pe- 
riphery of the ellipse is produced from two centers ; and 
by means of radii which have the property of increasing 
and decreasing proportionally. While the periphery of 
the circle is produced from one center, and by means of 
one radius, which always remains the same. Here is an 
identity of origin, and peculiarity of construction, which 
distioguish the circle and ellipse from all other curvilinear 
figures. 

When we examine the square and rhombus, we will 
see a similarity. The rhombus is nothing but a depressed 
square, having its sides equal and parallel, and all its an- 
gles equal to four right-angles. These are peculiarities of 
the square. ~ In: depressing the square to form the rhom- 
bus, the 2Je7'i?7ieUr is not altered, the sides are always equal 
and parallel ; and the angles are equal to four right-an- 
gles. The only thing changed by the transformation, id 
the area — with which Geometry has nothing to do ; since 
it belongs exclusively to the province of Arithmetic, and 
we are considerino: the 2,eometrical relation of magnitudes. 
When the science of Geometry discovers relations, then 
the science of Arithmetic may interpret them. Now, when 
we examine the peculiarities of the circle and ellipse, and 
consider the peculiarities of the square and rhombus : and 



22 • SO:\IE IM-OKMATION 



the iatimatc relations between the circle and square : the 
evidence bursts upon us, tbat the ellipse is nothing but u 
dejyrcsscd circle, 

If we form a square, with the transverse axis as a 
diagonal, and another with the conjugate axis as a diago- 
nal ; we wi!l find by computing the area of these squares, 
^ and the inscribed rhombus, that the rhombus will be the 

mean-proportional of the squares — thus : suppose the 

transverse axis is 8 — the conjugate axis will*be V 32 = 
5. 65G -|- ; the mean-proportional of the transverse axis, 
and one-half the transverse axis : then the area of the 
square formed on the transverse axis, will be 32 ; the area 
of the square formed on the conjugate will be 15.99516 -|- 
the area of the rhombus inscribed of the ellipse will be 22. 
624 -I"-; and we will have the proportion 32 : 22.624 -|- :: 
22.624 -I- : 15.99516 -|-. Now this is a property of' the 
ellipse, with circles formed on the axes as diameters. See 
Brande's Encyclopcedia, under Ellipse. 

Here is another coincidence of the figures — circles 
and ellipse, square and rhombus. In depressing the circle 
to form the ellipse, its periphery is the same ; but the area 
is reduced. As an exemplification of the above, we will 
draw a circle and inscribe a square ; then depressing them, 
we will have the ellipse and rhombus ; and by a simple 
.ji; lorocess, we can again have the circle and square. 

Then, having an ellipse, we can inscribe a rhombus ; 
and knowing the transverse axis, we can find the conjugare 
axis ; and from these determine the side of the rhombus : 
then forming from the side a square, which will be the in- 
scribed square of a circle, whose periphery will be equal 
[(;» to the periphery of the ellipse. From the inscribed square 

we can obtain the area and periphery of the circle ; and the 
area and periphv^ry of the ellipse. The area of the ellipse 
is the mean-proportional of the areas of the circles upon 
the transverse and conjugate axes. If we know the trans- 
verse axis only, get the mean-proportional of it and one- 
tt half of it; this will be the conjugate axis. The periphe- 
ry of the ellipse will be equal to the periphery of the cir- 
/ cle ; of which the side of the inscribed rhombus is the 

i mean proportional of ils diameter and radius. 



CONCERNING THE (J ROLE. 23 



Let the transverse axis be 12, then the mean-propor- 
tional of the transverse axis 12, and one half the trans- 
verse axis 6, will be the conjugate axis \^ 72= 8.485 -|-. 
One-half transverse axis squared plus one-half conjugate 
axis squared J will be the square of the side of the inscrib- 
ed rhombus; (Proposition XI., Book IV., Davies' Legen- 
dre) and the root of this square will be the length of the 

side of the rhombus, V^BG --{ \/'72\''oy ^36-1- 1*8 == 



('?) 



V 54. The square formed by the side of the inscribed 
rhombus^ is the mean-proportional of the square of the 
diameter, and the square of the radius of the circle whose 
periphery is equal to the periphery of the ellipse— thus : 
54 is the inscribed square of the circle, and the mean-pro- 
portional of the squares of its diameter and radius : hav- 
ing this mean-proportional, we can form the proportion — 

thus : 54 : 54 :; 54 : 2 (54) or 27 : 54 :: 54 : 108. Then 
2 

27 is the square of radius, or 5.19'-!- is the radius ; and 

108 is the square of diameter, or 10.39 -|- is the diameter. 

The square of radius, multiplied by three — thus : 27 X 3 

X 3 = : J i X 3 = 

■ ^ 2 ■ / 



,^'- t 


/ 


1 4 4 
\ ! 2 


2 


X 3 



r 



/ /12\2 _!-|/72;' \ 



o 



will be the area of the circle. Hence the square root of one- 
half the Slim of one-half the transverse axis squared^ and 
one-half conjugate axis sqi^ared^ ivill be the radius of the 
circle. The diameter multiplied by the ratio 3.4641 -|- of 
the circumference and diameter, will produce the periphe- 
ry of the circle and the periphery of the ellipse — thus : 

10.39-'-, X 3.4641,-:-= VTO8 X 3.4641 -|- = ^2(54) 



X3.464I,-i- = ^2(\54) X 3.4641,-'-= ^2('36-i-18) 
X 3.4641, -i- ^ ^'"^(/il^^I?) X 3.4641, -I- = 



4 



24 _. . _.:r>.iation 



^'^(M^r)' - - (vH;'^ y X 3.4641 -!-. Hence the square 

root of ticice the sum of one-half transverse axis squared^ 
and one-half cofijugate axis squared^ icill be the diameter 
cf the circle. Then, the periphery of the ellipse mill be 
tcual to the square root of twice the sum of one-half trans- 
verse axis squared^ and one-half conjugate axis scfiartd^ 
multiplied hy 3. 464 1 - -. 

The area of the ellipse, is the mean-proportional be- 
tween the areas of the circles, upon the transverse and 
coniuofate axes. Thus, the area of circ4e unon transverse 

sxls, is 3(-V-)^ ; and th^ area of circle upon conjugate axis is 
?( v'S)' ; then by completing the proportion, we have. 



3(W: V3(-^rx3(\:^)^:: ^ 3 (^/ x 3 ( v ,_.r : 3 
( <lJLf therefore ^3 (V)' X 3 (VtT)- is the area of the 

ellipse ; or, squa.re root of three times square of one- 
half transverse exis, imdtiplied by three times square of 
one -ha f conjugate axis^ will produce the area of the ellipse. 

These rules can be applied to the circle, which demoyi- 
strales clearly, that the circle and ellipse are, in principle, 
essentially the same ; and that the principle of tJte quad- 
rant is correct. Because, if there be an error in the be- 
ginning, that error would be multiplied by operations, and 
would be perceptible by illustrations- 

The radius of the circle is equal to the square root of 
onC'half the sum of one-half the transverse diameter 
squared^ and one- half the conjugate DrAMEiEPv scpxarcd. 

The periphery of the circle, is equal to the square 
root of twice the sum of one haJf transverse diametePc 
squared^ and one-hcdf conjugate diameter ^quared^ niul- 
lipliedhy 3.4641 -'-. 



CONCERNING TUB CIRCLE. 



The area of circle is equal to the square ro:^t of t J tree 
times square of one-half transverse diameter^ raultiplidhy 
three times square of one-lialf conjvgate diameter^ equal to 

THREE TIMES SQUARE OF RADIUS. 

To find the transverse axis when the diameter is known, 
we will find the mean-proportional of diameter and radius ; 
this will be the side of the inscribed square, equal to the 
side of the inscribed rhombus. The square of this side, 
will be equal to the sum of the squares of one -half trans- 
verse axis, and one-half conjugate axis. When we have 
the sum of the major term and the mean-proportional of 
a proportion, the square of the major term, when the ra- 
tio is 2, is double the square of the mean-proportional ; 
eonsequentlj, twro-thirds of the sum of the squares of one- 
half transverse axis, and §nehalf conjugate axis, will be 
the square of one-half transverse axis ; and the remaining 
one third will be the square of one-half conjugate axis- 
Let 10.39 -I- be diameter; then, 10.39 -!- : v^TO.39 ~|~~)r 
5.T9:T-. : : v^o 39 --!- X 5.19 -T- : 5.19 -I-. 



M039 -^1- X 5.19 -|- : 5.19 -j-. But 

squaring the terms of the proportion, we have 108 : 54 : : 
54 : 27. 

Ilence, 54 is the sum of the squares of one-half trans- 
verse axis, and one-haif conjugate axis, and V'54 = 
^^36-1- 18 = 6-1-4,242 ~|-, consequently, 6 is one-half 
transverse axis, and 4.242 -|- is one-half conjugate axis. 

When a demonstration in its natural progress arrives 
at a truth before demonstrated, even though its premiss be 
of such a nature tkat we cannot j^eadily acknowledge its 
truth, we are bound by the principles of the science, to 
admit the demonstration as being correct. Now at the 
start, we begun with a premiss, founded upon evidence] 
and in the natural course of the demonstration, we arrived 
at a truth, before demonstrated : (See Brande's Encyclo- 
paedia, under ^' Ellipse," where the area of the ellipse, is 
the mean proportional between the circles constructed on 
the transverse and conjugate axes.) and we found the con- 
clusion corroborating the premiss. What demonstration 



fi 



2b SOME INFCRMATICN 



in Geometry is more legitimate, clear and satisfactorjj than 
the demonstration of the curve line ? A premiss founded 
on evidence, arguments in their legitimate course arriving 
at a demonstrated truth, and a conclusion corroborating 
the premiss. Here are all the requisites of a complete 
Demonstration. 

Another property of the truth of science, or the ua- 
mistakable evidence of the natural order of things, is the 
consistency wliich a correct principle maintains throughout 
all its operations Upon this property rests the existence 
of science. Without it, science would degenerate to an 
art; and tho great researches of the mind, would dwindle 
down to mere inventions. And for these reasons^ we will 
dwell upon the consistency of the principle of the quad- 
rant. We will assume the principle as correct ; and test 
it with all the problems of the circle and it.- metamorpho- 
sis — the ellipse. By its means, when we know the radius, 
we can determine the area of the circle, and the length of 
any arc of the circle ; with the radius and arc, we ean find 
the area of any sector and segment; with the transverse 
axis known, we can determine the area of the ellipse and 
its periphery — the conjugate ax's, the periphery and area, 
and other parts of the circle, having a periphery equal to 
that of the ellipse ; also, the parts of the circles, construct- 
ed on the axes of tae ellipse. By this principle we can 
find the progression between the circle and the circum- 
scribed and inscribed squares, and the progression between 
the ellipse and the circumscribed and inscribed rhombuses. 
The latter progression is thus found to be similar to the 
former. With the circle and squares, let the radius be 4, 
then area of circle will be 48, that of circumscribed square 
64, inscribed square 3:2, the square of radius 16; and we 
have the progression 16, 32, 48, 64 — an arithmetical pro- 
gression ; and to find the progression between the ellipse 
and rhombuses : when the transverse axis is 12, the area of 
ellipse is 76.36 -|-; inscribed rhombus 50.91 -|- ; the 
figure corresponding to the square of radius — a rhombus, 
formed by one-half transverse axis, and one half conjugate 
axis, 25.45 -!-: and the circumscribed rhombus 101.82-1- 
— and we have the following progression : 

25.45 -1- 50 91 -I-, 76.36 -1-, 101.82 -|-. 



CONCKSMN«> THE CIRCLE. 27 



If we examine these progressious. we will see how they 
are formed — thus : the first. commeneiQg with the square 
of radius, we have that for the first term ; then adding to 
itself, we have the second teriii ; to itself twice, the third 
t6rm; and to itself three times, the fourth terai. The 
proportion of the ellipse, in a similar manner, taking the 
rhombus formed bv the square of radius for the first term : 
adding it to itself, for the second term : to itself twice, for 
the third term ; and lo itself three times, for the fourth 
term. TVhat is clearer than this, that the principle is con- 
sistent ? And beins' consistent, it is right. B\ this prin- 
ciple we can find the ratio ofcJ:he circumference to the di- 
ameter — which never was found before. Independent of 
every other consideration, its very use should recommend 
it. Bv the words of the immortal Bacon: — '^ The con- 

a- 

firmation of theories relies on the compact adaptation of 
their parts, by which, like those of an aich or dome, they 
mutuall}' sustain each other, and fonn a coherent whole/' 
[Hersghel.] 

It may be inquired, if the square E F G H, in Dia- 
gram No. I, d vides the distance between the circumscrib- 

ed and inscribed souares, whv is not I E F P equal to 

i ' •■■ ^ 

E A B F ■ This exnlains an important subject in mathe- 
matics— the difference between an ^-ithmetieal progression 
or proportion, and a geometrical progression or propor- 
tion. In a geometrical proportion and progression, we 
have a relation, which is derired from the nature of ratio : 
for instance, take in Dia:rram Xo. 1. the diameter, ^t)r its 
equal A B, and the radius or its equal M B. The arith- 
metical progression between them, if the diameter be 8. 
would be S. 6, 4. The geometrical proportion would be 

8 : ^ 32 : : ^32 : 4. Arithmetic beino; the science of num- 

bers, has reference to the relation of units : while geome- 
try has reference to the properties of lines. To the arith- 
metieal progression, we must use an arithmetical term — 
as twice, thrice. &c.; to the other, a geometrical term, as 
square, cube, &c. Thus, from the former, we will gQi 8. 
6,4,-16. 12. S— 24, 18. 12, &c.; from the latter, 8 : *''32 
:: v-32: 4—64 : 32 : : 3*^: 16, &c. In an arithmetical 
progression and proportiW. we have a relation which is d^ 
C - ^ 



28 SOME INFORMATION 



rived from the nature of difference. One is founded upon 
the principles of Geometry, the other is founded upon the 
principles of Arithmetic; one has reference to the dimen- 
sions of njagnitudos^ the other to the area or contents of 
magnitudes. Now, I E F P and E A B F, cannot be com- 
pared together, for the reason, that they are obtained 
through diifcrent principles. The square I P K L, is ob- 
tained by the geometrical principle of ratio. The square 
E F G H, by the arithmetical principle of difference. 
Hence, if the figures I E F P and E A B F, appear dis- 
similar, it is not because the square EFGH, does not di%'ide 
the distance between I P K L and A B C D ; but because 
the former square is formedvby principles diff^ent from 
the principles which form the latter squares. Suppose ^e 
form a circle of 48, and we wish to find the Sjuarc equal 
to it: the diameter will be 8, and -the radius 4, the mean- 

proportional ^o2 == 5.656 -'-. Having 8 and 5 656 -J- 
we cannot find the difference between them ; because 8 13 
iin arithmetical quantity, and 5.656 -'- is a geometrical 
quantity. But we must first find the difference between 8 
and 4, and then get the r^-^io between this difference, 6 and 
8. The square of this ratio will be the square equal to 
..the circle ; or, in other words, we will only reverse our op- 
erations in finding the arithmetical square - from our ope- 
rations in finding the geometrical square, which w>re first ' 
to find the ratio and then the difference. 

Theve is a property of the principle of the curve line, 
T^liich is characteristic and useful to an eminent degree. 
We can take a straight line, and make it the hypotenuse 
of a right-angled isosceles triangle. The vertex of this 
triangle will be the center of a circle, and either side will 
Tdc the radius. And this straight line above, will be the 
chord of the circle. The quadrant of this circle will ba 
equal to the chord and the altitude of the segment. This 
length will form the square equal to the circle. The chord 
will form the inscribed square of the circle. The chord 
and twice the altitude will form the circumscribed square. 
Upon these squares, .we can form a series ^f circles — in-* 
scribed and circumscribed ; and upon the circles a series 
.of squares — inscribed and cireuni^ribed ; and we can con- 



CONCERNIXG THE CIRCLE. 29 



tinue these series of circles and squares ad infaiitura, 
which each circle and square will be intimately related, 
one to the other. If we have a square, and ^ish to Ifave 
an equal circle : we will circumscribe a circle and inscribe 
a circle, and the intermediate circle between the circles, 
will be the circle equal to the square. This is a beautiful 
property of the principle of the curve line, and shows 
plainly and clearly the intimate relation between the 
straight line and the curve, and between the square and 
the circle. This property of the curve line, reminds us of 
the series formed bv the al2:ebfaic svmbols. and the arith- 
metical quantities. 

The facility of measurement is tiie grand object of 
geometrical science. It is not sufiicient to know the area 
of a surface, the contents of a solid, or the length of a 
line ; but it is a great desideratum to know the simplest* 
and most facilitious method for obtaining tho.^e things in 
ouestion. If we wish to know the length of a straight line, 
we have a unit of measure^ founded on the nature of the 
Uraight line for the 'purpose \ or, in other words, the unit 
of measure is the sar/ie in nature tL/id properties as the 
thing measured. The curve being essentially different 
from the straight line, we can not. by means of an unit of 
meo.sure founded on the nature of the straight line, ascer- 
tain the length of the curve. But we must endeavor, 
by some unit of measure^ founded on thz nature and 
properties of the curve^ to determine the length of the curve. 
For, let us try to find the length of the circumference by 
the diameter, we will have the circumference containing 
the diameter three times icith an indeterminate fraction. 
But if we have a straight line to be measured by a recti- 
lineal unit of measure, we would have no indeterminate 
fraction over. It is a truth in Geometry, that circumfer- 
ences are in proportion to their radii : this fact establishes 
two things — first, that there is a constant and invariable 
ratio between the circumferences and diameters ; and sec- 
ond, knowing the diameter, we can find the circumference. 
Circumferences being in proportion to their diameters, 
their lengths are dependent on the lengths of the diame- 
ters. Thus, if the diameters be 3 and 6, the circumfer- 
ence of the first diameter will be one-half the circumfer- 



^» 



0' SOME INFORMATION 



-cDce of the other ; if they be 5 and 7, one circumference 
will be two-fifths more, or two-sevenths less than the other. 
Here we have a rneam of measure for the curve line, and 
wc^havc only to adopt an uint of measure. Let the diam- 
eter be 1, or unity — th^n, we can call the circumference, 
thi? circumference of unity: any diameters — 2, 3, 4, 5, &c., 
v^^vrill have ciroumfcrences, 2, 3, 4, 5, &c., times greater 
than the circumferenee of unity. If the diameter is a frac- 
t'oii or an integer and a fraction, the same ratio will be 
maintained: for instance, if the diameter is -f- or 4 fj the 
circumference will be f , or 4 f of the circianferenre 
(/f iiuity. Suppose we have an arc of 60^ of the circum- 
ference whose diameter is 4. Now the circumference will 
be equal to 4 circuwferences of unity \ and the arc i of the 
circumference ; therefore, the arc will be equal to -J- of 4 
drcuhiferenees of unity ^ or ^ circumference of unity. This 
circumference of tinity may be one inch, one foot, oneyardj 
or due mile, it \yill depend upon what we wish to measure;" 
and it will ])e familiarized by frequent use. By its adop- 
tloL, we will be rid of the disadvantages and inaccuracies 
attending the use of decimals ; and it will be more roT^s^^t- 
ent with the principles of the science. 

In Diagram No. 2, wc have a circumscribed and au 
in>:cribed square. The circle is evidently between them , 
tthe circumference of this circle is a regular curve line, be- 
ing produced by a radius from a point within, called the 
centre. We perceive, when the circumscribed and inscrib- 
ed squares are drawn, as by Diagram No. 1, that the cir- 
cumference will touch the middle points in the sides of the 
circumscribed square ; and the extreme points in the sides 
of the inscribed square.' And that, in whatever position 
v/e may place these squares, this fact will be made mani- 
fest* which is a remarkable coincidence — one that signifies 
more thayi nothing. There is another coincidence in the 
quadrant of a circle and the side of a square. Now these 
squares are perfect, and so is the circle. There is evident- 
ly a relation between them, and this relation must ezist 
for some purpose. TVe perceive, also, that no straight lines 
touching the above points, can divide the distance between 
the squares. And it will be to no purpose to argue that 



y 



CONCERNING THE CIRCLE. 31 



irregular curve lines will do it ; because, there must be 
^xeason in every argument, and no single line i)ut a regular 
curve line can touch these points. Four times square of 
radius produce the area of the circumscribed square ; twice 
square of radius produce the area of tbe inscribed pquare \ 
and three times square of radius produce the area of the 
circle. The circumscribed square is the square of the di- 
ameter; and the inscribed square is the square of the mean- 
proportional of the diayueter and radius. Now, when all 
:hese facts are duly considered, and our mind unprejudiced 
turned to their meaning, we can naturally inquire : What 
do these relations signify ? If we adopt the evidence that 
the circle is intermediate between the squares, we will have 
discovered a principle, ^and in its m?:nifestations, beauti- 
ful in its deductions, and useful in its operations But if 
it be rejected, because it is an undemonstrated truth, Ge- 
ometry must be deprived of a jewel rich for its casket, but 
unappreciated. And Geometry must still remain a secon- 
*dary science — borrowing the lights of other sciences, in- 
stead of shining brilliantly with its own, A principle is 
undemonstrable ; it is inherent, indefinable, intangible, and 
"unseen. It is that governing impulse which stimulates the 
exertions and conceptions of honor, truth and virtue. It 
is that hidden property which gives power and force to 
steam. It is that incontrovertible and undeniable fact 
which gives conviction and proof to mathematical demon- 
strations. It is that something, but we do not know whatj 
which controls our faculties, and forces our reason to sub- 
mission. 

It is a great requisite, and important desideratum, that 
every proposition enunciated, must have a premiss before 
satisfactorily demonstrated ; or must be of itself evidently 
true ; or,.it^musfe lead by plain and undeniable demonstra- 
tions to a truth before demonstrated or evidently known, 
before we can acknowledge its truth, applicability or just- 
ness. And in the investigations of geometrical subjects, 
it is this fact which gives its demonstrations permanency 
and reliability — without which, the plainest and grandest 
truths, of Geometry would be as insecure as the fancies and 
phantasies of the mid-night dreamer. 



^^ SOME INFORMASiON 



\\ neu we examine the science of Geometry, we will 
fand that It IS founded on the principle of the straight line. 
Ail Its problems are solved by means of its properties The 
great science of Geometry is founded on the principle, that 
It IS the shortest line between two given points. Geome- 
ters place this fact among the axion^s of Geometry, ag a 
truth which cannot be made more evident by a demonstra- 
tion. Assuming this as a truth, they have been enabled to 
calculate the various distances upon the .■surface of the 
earth ; measure the dimensions of substances ; in a word 
to make discoveries without name and bevond number It 
18 a principle which guides us in all ourValks of life'- it 
IS mtimately associated with our being, achievements and 
necessities A straight line is one that maintains the same 
direction between any two of its points. Althou<^h this 
defines what a straight Kne is, it does not prove it to be the 
shortest line between the points. It is an assumption 
founded on the evidences of our senses ; which, as far as 
we are able to judge, we are obliged, from the constitution 
ot our minds, to consider as true and incontrovertible. In 
none ot the demonstrations, theorems, problems, or lemmas 
ot geometry, do we find any knowledge of tke principle of 
the curve line. Even in that part which treats of circle- 
we perceive relations between the straight line and the 

?;«?') Tl^ ™"', T 1^^^°^^^%° of the'principle .of the 
.tra ght hue : and this knowledge affords no insight of the 
mysteries ot the curve. And when we leave geometry 
and go to Trigonometry, we are not any wiser oi' the curve 
we attain here a knowledge of the sines, co-sines, tan<rent.' 
co-tangent., &c of the angles. We also find the values 
ottjie angles, which i.s not the lengths of the arcs, but the 
^jMce mcludea between two straight lines meeting at a 
common point. These are all straight lines, or the proper- 
ties ot straight hnes-not curves, Sor in any way divK- 
|^£ the principle of the curve. The arcs are used merefy 
yr the measurement of the angles. The inherent propei^ 
ties ot the arc. can be considered in the same bVht as any 
o.her means of measure : for instance, they may bcrec-ard- 
liVl T'^, ^' ^'^^''^ of any yard-stick, in measurinl the 
v/'lf 't\'' °'', '"^ ''^''°' ^° measuring distances on 
tfceed.tb. The only property or inherent value display- 



CONCEE.NING THE CIRCLE. 33 



ed here, is the use. Trigonometry is divid«d into two 
parts, called Plane and Spherical Trigonometry. In Spher- 
ical Trigonometry we have the same quantities, and their 
treatment here does not enlighten our idea of the principle 
of tfee curve line. Sines, co-sines^ tangents, &c., also an- 
gles are employed here ; but they all receive their values 
from the properties of the straight line ; and the treatment 
of the angles here, is of the angles formed by imaf-piary 
straigkt tines. The only diiference between Plane and 
Spherical Trigonometry is, that in the former, these differ- 
ent quantities are employed to find the lengths of the sines, 
co-sines, &q., and the angles ; while in the latter, the same 
quantities are used to find the angles exclusively. In the 
former^ straight lines are used, and the curve lines wider- 
stood ; in the latter, curve lines are used, and the straight 
lines understood. They are essentially the same, founded" 
upon same principles ; with the difierence only of applica- 
tion and operation. They both owe their existence to the 
relations between the straight line and the curve ; and the 
principle of the straight line. 

The Differential and Integral Calculus, is that branch 
of Mathematics, which considers the difference of variable 
Quantities reduced to the infinitesimal ; and treats of the 
integer or sum of these infinitesimals. Before we discuss 
the object of this branch of mathematics, we will enter 
into an eyamination of its principles* The principles of 
this branch of mathematics^ is founded upon the conception 
of a difference between two infinitesimal variable quanti- 
ties. Now, in the first pMce, these infinitesimals have no 
physical existence. They are imaginary quantities; for 
they are reduced so infinitely small, that they have exist- 
ence only in the imagination of the mind. Hence, we see, 
that the principles of this part of mathematics, are found- 
ed upon the relations of imaginary quantities; therefore^ 
they have no more real existence, than the quantities upon 
which they are founded. In the second place, our concep- 
tion of mathematics is. that it treats of abstract quaQtities, 
such as units, symbols and lines ; and though these quan- 
tities are abstract, vet thev have real existences. We do 
not mean, because these principles have no .real existence, 
that they are untrue— for we are not of that skeptical 
turn of mind, which believes only in tbino-s we touch, see. 



SOME INFCR^v^ATION 



:i< 



w 



hear, smell or taste. But we wish to intjuire into the proprie- 
Ij cf applying principles found^ed upon imaginary tluDgs. to 
tLings which have real existences. It may be said, that no 
principle has a real existence ; that it is inherent. We 
wish such to draw a difference between a principle estab- 
Jished upon some well known and admitted fact, and one 
established upon imaginary existences, So much for the 
principles. 

The object of the Differential Calculus is, to find the 
differentials of, or various increments to, variable quanti- 
ties. These differentials are the infinitely small quantities 

^affixed to variable quantities of assumed values. When 
a quantity is thus increased, it passes to a new form and 
TpJue— and we can increase this quantity at pleasure. The 
Differential Calculus, considers those increments, and de- 
termines the assignable value of each. The object of the 
Integral Calculus is, to integrate these differentials to the 
function from which they were obtained 

Our purpose is the examination of the prmciples of 
the Differential and Integral Calculus, in their applica- 
tion to the curve. The Differential Calculus considers 
the curve a polygon, of an infinite number of sides. It 
suppos3S these sides to be reduced so infinitely small, that 
,lhey become the hypotenuses of right-angled triangles^ 
formed by what are termed the differentials of the eo ordi- 
xates of the points of the curve. The theory founded upon 
this supposition, will agree with the polygon of infinitely 

' small sides. But when we exai«:ne the nature of the curve, 
we will see that this theory is perfectly absurd, when ap- 
plied to it. The side of a polygon, no matter how infiait«- 
ly small we reduce it, will still be a straight line ; all the 
points of which will be in the same direction, while those 
of the curve coutinually change their direction. Let us 
conceive the lines reduced to two points each ; the direc- 
tions which each two points will assume, will distinguish 
the characters of the lines Now, the questions arise : as 
the curve is so different from the strai^rht line, can the 
same theory be applied to both ? And if not, to which 
on.e does it properly belong I In the first place, the curve 
and straight line form different orders of magnitudes, and 
each order has properties peculiar to itself: certain rules 



CONCEaNlNG THE CIRCLE. 35 



and principles have been f :)unded upon these propertie,?? 
distinct and characteristic. No rule or principle of one 
order can be applied to the other. In the treatment of the 
various problems of each order, our operations are con- 
formable to their respective properties. Then, as the 
straight line and the curve are so different in their natures^ 
properties and principles, the answer to the first interrog- 
atory, is evidently, No ! In the next place, the theory is 
founded upon the properties of the polygon ; and the poly- 
gon belongs to the order of magnitudes formed by the 
straight line : therefore, the theory properly belongs to the 
straight line. 

The Integral Calculus endeavors, by the principles, of 
the Differential Calculus, to effect the rectification and 
quadrature of the curve. The Integral Calculus can de- 
termine the length, or any portion of the length, of the per- 
injeter of the polygon, or the area or any part of the area 
of the polygon : but to effect the rectification or o[uadra- 
ture of the cui've, would be utterly impossible. 

The series: XT^s^T^ = « "h il7<^ -'- iZIT^ 

-'- ;r7T""T~~7 -!- &c., &c- &C;., sufficiently developed niak- 
•' 2 4.6. 7 a^ : , . ■ ' 

ing a == :/; = 1 ; would produce the arithmetical quantiij* 
I.57079632o7048966i923 -j-; and this quantil^y ^^ ^^^ 
length of that portion of the perimeter of the polygon of 
an infinite number of sides, inscribed in the quadrant of 
the circle ichose radius is unity. 

The Differential and Integral Calculus was used ira- 
mediately upon its invention, for the purpose of finding 
the length of the circumference, and the area of the circle. 
It was thought sufficient to know the former in order to 
obtain the latter. The series above stated, was considered 
the desired length, for the quadrant of the circle whose 
radius is unity : this gives the i:atio of the circumference 
to the diameter ; as the quantity 3.1415926 -;-, or in gen- 
eral computation, 3. 1416 . With this quantity as the ratio 
of the circumference to l^he diameter, the Trigonometrical 
tables of the natural sines, co-sines, tangents, co-tan2:ents, 



3G 



OMt: INFORMATIOX 



ifcc; and the logarithmic sines, co-sines, tangeais. &o , 
have been computed. The amount of labor bestowed upon 
this ratio as the correct one, has Vjeen immena^. And it 
IS with feelings of the greatest reluctance, that we have 
undertaken to combat so much intellectual labor, perse- 
verance, zeal and researches. But it would be recreant 
to the high trust reposed in us, to hesitate upon the con- 
summation of the duty imposed upon us by the giits of 
understanding, judgment and reason; on account ^of the 
arduousness and responsibility attendiu^c the task of cor- 
recting the errors of a science supported by t"he greatest 
geniuses, and brightesr intellects., which have ever adorned 
the human race. What we have undertaken, has been 
prompt-d by the spirit of conviction, and with motives of 
integrity. In our researches, we have met with obstacles 
in our path, and we have engaged in the dutv to remove 
them. If we do not receive that acknowledgement of our 
services immediately, it will not disappoint our expecta- 
tions, nor cause a murmur to arise because of the seeminr*- 
neglect. For we are well aware of the difficulties of im"^ 
mediate judgment, and are fully prepared to make allow- 
ance for the opposition of prejudice. It would, indeed 
be remarkable if the wheels of revolution should move 
smoothly on without some clogs to impede its procuress, 
and retard its motion. We well know the ridicules which 
wcfe lavished upon Bacon by his countrymen, when for- 
eigBcrs were appreciating his genius and talents 

Mathematicians give 3.U159-2G - -, as the area of 
,the inscribed polygon; and the same quantitv as the area 
ot the circumscribed polygon, of the circle whose radius 
IS unity— the polygons having 62768 sides each. We tave 
not undertaken to develop the decimals to infinity^ nor to 
carry on the computations to the limit which is necessary 
to obtain the accurate results, as it would be superfluous, 
and altogether foreign to the matter we have under con- 
sideration. ^ Even if our intellectual labors did succeed to 
obtain the infinite range of the decimals, and did compute 
the number of sides of the polv^ons ad innmtum%^ 
would then only know the areas^^and perimeters of the 
polygons—and no stretch of the imagination could believe 
them tne area and circumfereD.ee of 'the circle. But one 



CONCERNING THE CIRCLE 67 



thing we know, that the area and peririietcr of the circuro. 
scribed polygon, no matter how often we increase the num- 
ber of its sides^ must continue to be, and will alwajs re- 
main^, greater than the area and circumference of the circle. 
And the area and perimeter of the inscribed polygon less. 
Mathematicians, however, have made them Ct^^ual ; but by 
what reason, we ple^d ignorant. 

The true ratio of the circumference to the diainetar 
is, 3.4641016150 -|-. The operations and computations 
are open to the inspection, correction and judgment of the 
whele scientific world. And we are proud to say, that 
there is no analysis too minute^ nor, no criticism too 
severe, to which we will not gladly, willingly and cheer- 
fully submit them. 

The circumference of the circle is divided into 360^: 
each degree into 60^/ and each minute into 60^^; making 
in all 1296000 equal parts or seconds. This measurement 
is altbo^ether arbitrary : and it was adonted bv the ancient 
Greeks. The French mathematicians have endeavored to 
substitute the decimal measurement, by dividing the quad- 
rant into 100 equal parts, but were unsuccessful, owing to 
various reasons. Since the Trigonometrical tables will 
have to be revised, we will use the decimal system in the 
division of the quadrant, and compute the values of the 
angles accordingly. 

Let the radius be one ; then the semi- circumference 
will be 3.4641016150 -j-, and the quadrant L7320503075 
-]- ; therefore, each part of the quadrant is equal to 

lli!!^!^— = 0.017320508075 -.-. This 0.017320^ 

100 ^ 
50^075 -h, is an arc: and this arc has a sine/co-sfee, tan 
gent, co-tangent, secant, co-secant and versed sine. These 
lines have each a value dependent upon the length of the 
arc. We can call the quadrant aa arc of 100^, read one 
hundred degrees : then 0.017320508075-;- is the length 
of 1 ^ ; each de^jree divided into ten equal parts or min- 
utes, would be 0.0017320508075 -j-; and each minute in- 
to ten equal parts or seconds, would be 0.00017320503075 
-1~ ; using the Srst ten figures, or .0001732051, will be the 
sine of 1^^ In Trigonometry , we have C5S,= in sine^) 



■^' SOMR INPORMATIOV 



oonsequentlj the cojiuv- of 1 '' = .Viyyyy<j935 }^^y^ J^ ff»-» 

= 0.000] 7320 - -. cot. = '±*: = =177 .. i ; I -.0 ; T' 

Ihese quantities are the natural s\m, cosine, tangeat. co- 
tangent, &e, &c., of the angle of 1 '/. We will not dwell 
more on the values for arcs of greater extent in the pres- 
ent place. At another time, we propose to arrange the 
tables of natural sines, cosines, &c., and tlie logarithmic 
smes, coMnes, &c. ^\ e have adopted the decimal system 
for the measurement of the quadraut, for the reason tha* 
It IS morereadily applicable and natural to the scient'-fir 
inTcstigation ; and for the purpose of making one step to'- 
wards the introduction of an universal system of nunieri 
cal computations. This latter consideration is of immense 
utihtj- to the sciences. As their principles and demon- 
strations are universal truths, it is quite requisite that the 
language through which they are promulgated, should b,^ 
one of ea.«y and uuiversa] internretatioa. 



CONCERNING THE CIRCLE. 39 



A DISCUSSION OF THE PROPERTIES OF THE 
STRAIGHT LINE AND THE CURVE. 



According to the views entertained by Mathematicians, 
the Rertification of the Curve is the determination of the 
length cf the straight /z'/zs equal' to the curve. And the 
Quadrature of the Gurve^ is the determination of the 
surface of the square^ equal in area to the space bounded 
in part or entirely by that curve. The problems are con- 
sidered resolved, when an expression is obtained denoting 
the area and length in question, although this expression 
is, in fact, nothing but an apprcximation. [See Oourte- 
nay's Calculus and Brande's Encyclopaedia ] Our pur- 
pose is simply to examine the above definitions; to see 
how far mathematicians are consistent with truth and sci- 
ence ; and to give candidly our opinion of their efforts. 
Antiquity is fall of power and influence. Age penetrates 
deeply its roots, and obtains thereby, a firm hold. Vene- 
ration is the noblest of emotions ; when given full scope, 
it becomes the mightiest ; and, like other things, when in- | 
dulged discretionally, it is a virtue, but when otherwise, 
it is a vice. The Sun which shines so brightly, when ex- 
amined closely will be found to be full of spots. And is 
it strange, that the thoughts of men should be erroneous ? 
The science of Geometry treats of lines ; Arithmetic of 
numbers ; and Algebra of symbols in their applications to 
quantities. We thus see distinctions between the sciences. 
We can apply Algebra and Arithmetic to Geometrical pro- 
blems; but they cannot illustrate a Geometrical principle. 
Because, they would then be illustrating an algebraic or 
arithmetical principle, not a geometrical one ; as, in either 
case, they would be producing an algebraic or arithmetical 
result.. Mathematicians have heretofore endeavored, by 
the use of the numbers of arithmetic, and the symbols of 
algebra, to effect the rectification and quadrature of the 
curve. A moment's reflection will convince them of the - 
absurdity. We can express an arithmetical quantity by 
numbers, an algebraic quantity by symbols ; but can we 
express legitimately or scientificj^lly a geometrical quantity 



40 SOME INFORMATION 



by either? What right have numbers or symbols to be 
used in geometry ? Geometr}^ is a pure science., a legiti- 
mate one. Its principles are fundamental truths, estab- 
lished by reason and nature. It is less arbitrary than the 
other mathematics. It is obligated to no science ; but 
other sciences are obligated to it. 

ARCHiMEDEg considered the circumference of the circle 
the same as the perimeter of a polygon. * Other mathema- 
ticians engrafted this idea in their miuds in all their en- 
deavors to rectify and quadrafy the curve ; in other words, 

. they regarded the curve the tame in nature as the straight 
line. Volumes have been written, years have been spent, 
upon this theory. Minds such as Newton, Leibnitz, and 
others J have associated this idea with great intellectual 
achievements. The Differential and Integral Calculus 
received vitality and strength from its strong re- 
semblance to truth. And so great was the belief in this 
idea, that it came to be established as a fundamental prin- 
ciple of geometry. Hence, the definitions— the rectifica- 
tion of the curve is the determination of a straight line 
equal in length to the curve ; and the quadrature of the 
curve is the deter iui nation of the square equal in are'i to 
the space bounded by the curva Mathematicians claim 
that to have found expressions denoting the objects in ques- 
tion satisfy the problems. We wish to call attention to 
the words '^ determination-^' and " expression.^' These 
words, in their present place, have great significance. De- 
termination may be applied to either algebra or arithme- 
tic; for instance, we can determine the root of an equation, 
or the product of numbers; but we can not use it in a 
geometrical sense without associating an arithmetical idea 
with it. The same remark may be made in regard to the 
word *^ expression.^' We can measure oxdraio a line, but 
we can not express a line with geometrical propriety. But 
the word may be applied to algebra or arithmetic without 
any restrictions. We have dwelt upon the words, for the 

• purpose of showing a source of error in the treatment of 
the geometrical problems concerning the circle. 

Not only have mathematicians confounded the curve with 
the straight line; but in their idea of its solutions, they 
have confused their minds. with improper and unscientific 



CONCERMXG THE CIRCLE. 41 



terms. Science is based upon the natural o?der of things. 
Great preciseness and caution are required for the devel- 
opment, of scientifio fiicts, since truth is so timid and re- 
cluse, that unless we use proper means for its attainment, 
our efforts will fail of success. The slightest error in our 
mathematical operations will produce a wrong result ; so 
that, to obtain correctness, we should clearly understand 
and rightly interpret the requirements of the question. 
Blinded with a wrons: idea, mathematicians have tried 
every expediency to solve the problems with like success, 
until finally they have struck upon the novel and conveni- 
ent plan of *' expressing " the solutions. Are any of the 
solutions to the many problems in Geometry, mere expres- 
sions ? Are they not logical conclusions ? An expres- 
sion may be logical or not, but a conclusion presupposes 
the acme of logic. Geometry is, in some* degree, mechan- 
ical — hence we speak of the solution not the determina- 
tion of a problem. 

Although there is an essential difference between the 
straight line and the curve, yet we find in the progress of 
the science, some intimate relations between 'them. For 
instance, we perceive that regular and similar polygons 
have their perimeters proportionate to their apothems ; 
and their areas proportionate to the squares of their apo- 
thems. Circles have their circumferences proportionate 
to their radii, and their areas proportionate to the squares 
of their radii. Now, this coincidence between regular and 
similar poljgODS and circles, would appear at a glance, that 
they are similar, and have corresponding properties : that 
principles founded on the one, should be adapted to the 
other. And in fact, at a glance, they do appear one and 
the same. But with a little reflection, we will perceive 
distinctions between them. The polygon and circle are 
geometrical magnitudes ; their properties come under the 
cognizance of the science. The straight line and the curve 
line are geometrical quantities. They are of the same 
genus ; but different species, They have intimate rela- 
tions; biit characteristic dift^erences. They are both consid- 
ered as being made of a certain number of points — those 
maintaining the same direction, constituting the straight 
line ; those continually changing their direction, congtitiit- 



42 SOME INFORM \TION 



tiog the curve line. They are both lines, and as lines their 
relations are intimate ; but being different kinds or char- 
acters of lines, their properties are distinct. Again, reg- 
ular and similar polygons ordy^ have their perimeters and 
apothems proportionate. These polygons are composed by 
a certain number of similar triangles. These triangles 
have their homologous sides proportionate. Although 
triangles arc polygcns ; yet under all circumstances they 
are not regarded as such. The properties just observed, 
are dependent on the triangles ; they are demonstrated by 
the triangleSj not by the polygons, which comprise the tri- 
angles. Wo make this distinction, because geometers in 
treating of triangles, draw a line between them and poly- 
gons. The properties in question, do not belo^^g to the 
polygons as such ; but are derived from the nature of tri- 
angles. Hence we observe, that when regular polygons 
comprise a number of similar triangles, the properties of 
those triangles a^e deducible to the polygons, ^nd not 
otherwise. Therefore, we see whence these properties of 
regular and similar polygons are derived. Now, let us 
examine the circle. The formation of the circle is well 
known — wh ch shows plainly the connection between t .e 
ciicumference and radius. Also the fact, that there is an 
invariable ratio between the circumference and radius, 
establishes at once, the proportion of circumferences ani 
radiij and not from any ol the properties o-f the triangles. 
Although there are propoi'tions between the perime- 
ters and apothems of polygons, and the circumferences and 
radii of circles : yet, it is not because they have similar pro- 
perties, but because, being geometrical magnitudes, and hav- 
ing the intimate relation of lines, they have certain connec- 
tions and agreements. In regard to the areas of polygons 
being proportionate to the squares of their apothems — 
this proportion is derived from the property of similar 
triangles being proportionate to the squares of their 
homologous sides. The apothems of the polygons are the 
altitudes of the triangles. The altitudes of the triangles 
and homologous sides of the triangles ; therefore, the areas 
of regular and similar polj'gons are proportionate to tha 
squares of the apothems. 

The area of the oircle is produced by the product of the 



>*....'„- ■» 



CONCERNING THE CIPvCLE. 43 



square of racliii^, multiplied by three. Is o^r^ three being a 
constant factor in the production of the area of any circle, 
it follows very clearly, that when areas of circles are com- 
pared, that the squares of their radii must maintain the 
>same proportion^ even from the quality of relation, or 
from the property of factors ; not from any of the proper- 
ties of triangles. 

PolygoBS and circles^ being geometrical magnitudes ; 
the straiccht lines and curves bein^ g-eometrical quantities: 
the relations of su(5^b' magnitudes and quantities, are com- 
mon to the polygons and circles, but nothing. more« The 
straight line and the curve line, compose different geomet- 
rical magnitudes. The straight line composes the polygon, 
and the curve the circle. The polygon and circle being 
different geometrical magnitudes ; therefore, they must 
have the peculiar properties which distinguish such magni- 
tudes. The polygon receives its properties from the tri- 
angle; the triangle from the angle* and the angle from 
the straight line. The circlevreeeives its properties from 
the curve line. The properties cf the straight" line are the 
fundamental principles of Geometry. The principles of 
Geometry are assumptions. The properties of the curve 
line, must be established upon the nature of the curve — 
this nature must be peculiar to the curve ; and being pe- 
culiar tcf the curve, it must be a principle cf the curve. 
A priociple i^ undemonstrable ; it must be assumed, and 
this assumntion must be fouaded udou evidence. The 
principle of the straight line is, that it is the shortest line 
between two given points. The principle of the curve 
line is, that the quadrant of the circle is equal to the sum 

of the base or chord of the se<rment, aud the altitude of 

... 
the segm.ent. For instance, the strai^hc line distance be- 
tween the extreme points of th qu^.drant, is the chord of 
the se<]:ment ; and the curve line distance is the arc of the 
segment — equal to the chord of the segment plus the alti- 
tude of the segment. The altitude being the variation 
between the chord and the arc. 

The principles of the straight line and the curve are 
so different, that we cannot establish any truths dependent 
on the one, by any demonstration based on the principle 
cf the other. As we before remarked, they being geomet- 



44 



SOME INFORMATION 



rical quantities, have relatious as such; but they have 
properties peeu^liar to their kind. It would be absurd as 
well as useless, to demonstrate the properties of the curve 
from those of the straight line; in other words, we must 
demonstrate the curve line by the curve line. But before 
we can carry on our demonstrations, we must have a 
premiss. It would be much easier for our demon6tratk)ns 
(when we say easier, we mean to receive the approbation 
of others,) if this premiss has been a]ri3ady demonstrated : 
but we know that before any thing can be done, there must 
always be something given or assumed, to commence upon • 
otherwise, we can accomplish nothing. This something 
given or assumed, is not impracticable to the establishment 
of truth ; for, in fact, truth is in no way established, but 
through the means of assumptions. Even th% existence 
of God, cannot be proven without an assumption. This 
assumption is the suggestion of the human intellect. And 
when an assumption is thus founded, its truth can be made 
evident, by a method of reasoning termed the a posteriori. 
When we look around us and see the effects of goodness* 
intelligence and power; and above us, and seethe same— 
the mind, instinctively, suggests causes for these effects. 
And when we perceive an identity of purpose, and au 
unity of design; we must believe these ejects to sprincr 
from one common cause; and this, the great Cause which 
we declare God. Before thi.s conclusion, our intellect su^*-- 
gested the relation of cause and effect—that where there 
is an e&ct, there must be a cause for it. This first pro- 
position or premiss, which ever we may call it, is not de- 
monstrable; but it is assumed, from the natural order of 
things. Our mind is peculiarly constituted for the per- 
ception of agreements and diiferences. We perceive in 
the order of nature, things answering some design ; we 
trace in them relations ; we find in them a ready adapted- 
ness for some purpose; we at once infer some principle 
controlling them to a particular end. This is cause and 
effect; m the various existences around us, we perceive 
certain relations, certain designs and certain purposes, all 
corresponding to particular ends. Here we find ao-ree. 
merits, founded upon the natural order of things. H°ence 
we mler from these agreements, some controlling principle; 



COXCERXI^-Q the: CIPcCLE. 45 



tbat there are in all things strong resemblances ; similar 
elements of composition ; certain common, laws goyerning 
these elements — shewing a sameness in their natures : and 
consequently identity in the means of origination. Xow 
our intellect sucrorests that nothinir originates of itself; that 
there must be an originator — a cause for the effect. Hence 
we assume for every effect there must be a cause. And 
upon this assumption, founded on the cvmences around 
us. we declare there is a God. 

VTith regard to the principle of tbe straight line, it is 
an assumption; but owing to the evidence of our senses, 
and to its consistency throughout the whole extent of Ge- 
ometry, we declare it to be true. Although this proposi- 
tion of the straight line is not susceptible of direct demon- 
stration or reasoning a^ri(;ri; still its truth is not in any 
way affejted by the a posteriori method. 

The same can be said of the principle of the curve 
line. Vre see with our eyes, that the arc of the quadrant 
varies from the chord, the distance of the altitude of the 
segment. And it is also very palpable, that the arc when 
straightened, must extend the distance of the altitude be- 
yond the extremity of the chord. From the relations of 
the circle and squares, circumscribed and inscribed ; from 
the consistency of this principle with all the requirements 
of the problems, regarding the circle and ellipse : and last,, 
though not least, from the face that there must be a prin- 
ciple peculiar to the curve, as well as one peculiar to the 
straight line :• and that this principle, whatever it is, can- 
not do otherwise than correspond with the principle above, 
in its consistency and results ; and that it must be assum- 
ed, becau-se, being a principle, it is primary to demonstra- 
tion ; for without some principle or assumed truth, demon- 
stration cannot proceed. "We must conclude that the above 
principle, having all the qualities and properties of the 
right one — that it is, and must be, right-. This being the 
principle of the curve line : it is a truth of the natural 
order of thiy^^s, not of argumentation. We must consid- 
er it as such ; bv its means we can advance the state cf the 
science more than any sup]7<rition can do. 

"We have pursued these remarks thus far, to show 
that the evidences of o#r senses, when controlled by ma- 



46 SOME INFORMATION 



turs reflection and correct iucl foment, never lead us wroTio: : 
and that, if we persistently cling to tlic extreme idea of 
Geometry — to take nothing for granted, but what is de- 
monstrated to be true — then we must obliterate from the 
annals of science, the truths of Geometry and its kindred 
sciences. 

In Geometry, all the problems are solved, and all the 
demonstrations carried on, by the princi[le of the straight 
line. And no proposition will be admitted, unless it be 
established on truths devolving on the principle of the 
straight line — since the whofo system of Geometry is 
founded upon this principle. And when any problem of 
this order is to be* solved, theie is alway some axiom or 
demonstratiouj either remotely cr intimately connected 
with its demonstration, which will render the path to its 
solution direct, easy and plain. There will always be 
some link, either at the beo;innin£;, or in the middle of the 
chain of its arguments, which Y\dll connect it with some 
alreiidy established or admitted truth. And when the 
arguments for the solution of any problem, do not lead to, 
nor recognize, the principle of this system, it is evidently 
plain, that to whatever conclusion thes^e arguments may 
lead, it will be inconsistent with truths devolving upon 
this system as a science. But when we have a problem of 
a nature essentially diiFerent from the problems of this syj^.- 
tem, we must be guided by principles essentially different. 
Because truths established upon one principle, are absurd 
when applied to another. The straight line and the curve 
line are essentially different. They have nothing in com- 
mon ; and no truths founded on the one, can be applied to 
the other. Yv'h^n we have problems of the nature of the 
curve line, we cannot solv^e them by rules governing the 
straight line; nor can we even use a truth demonstrated 
by the straight line, for the demonstration of a truth de- 
pendent on the curve. Our premises cannot be fpunded 
upon evidences, nor demonstrations of the straight line; 
but they must be established upon principles exclusively 
of the curve And for the quadrature of the circle, we 
cannot find in the whole range of the geometrical science, 
om truth to assist us in the demonstration of the princi- 
ple of the curve line. By the science of Geometry, in 



C0^'CERN1NG THE CIRCLE. 47 



the absence of demonstration, we appeal to evidence ; there- 
fore, consistent with its great principles, and for the estab- 
lishment of grand and iniportant truths, when we arrive 
at that point where demonstration becomes impossible, we 
must resort to evidence. 

It may be obligatory on our part, since we have ap- 
pealed so much to evidence, to examine its merits and 
show its imnortance. Evidence has all xhe povjer of de- 
moDstration. without the chain of its aro-uments. Evi- 
dence is inherent, indisputable, perfect and eternal : de» 
mcnstration receives its reliability and permanence from 
the links of its arguments; and when one is fractured or 
destroyed, demonstration loses its strength and virtue. 
Evidence is accepted on account of'the constitution of our 
minds ; demonstration on account of the ingenuity and 
sophistry of the logician. Evidence appeals to higher 
grounds than the perfect arrangement of words ; demon- 
stration receives its support from the facility and adapta- 
tion of lan^uao-e. Thouo:h we consider evidence thus fa- 
vorably, we would not disparage demonstration ; for, al- 
though we owe the principles of Geometry to evidence, 
we owe the scietice of Geometry to demonstration. By the 
merits of demonstration, we perceive the application of 
those principles. It is the part cf demonstration, to dis- 
cover the hidden truths and unfold the secrets of facts. 
It is the virtue of demonstration, that it can trace through 
the labyrinths of mysteries, the path to success. Evidence 
is a demonstration, and demonstration is an evidence; and 
it is to the permanency, reliability and success cf a science, 
that we can use such powerful auxiliaries 



48 SOxAlE INFORMATION 



CRITICAL EXAMINATION OF THE ALGEBRAIC 

ANALYSIS, 

Kuowledge is progressive. When we compare the 
extent of the modern sciences, with the scanty acquisitions 
of the ancients — the great improvements and advancement 
of modern researches, astound the mind more than all the 
astrological signs and magical sways of the oracles, and 
other depositories of ancient lore. The mind of man 
seems in that period of mental poverty, to be busied with 
materiality, and to clothe itself with the uncouth garb of 
superstition and dread. Its movements were attended with 
a kind of cautiousness and distrust. It dealt in theories 
impracticable, and speculations unrealizable. It was not 
until facts had been multiplied, and the genius of man 
awoke and discovered tae orders of relations, the traces of 
resemblances, the points of contrasts, and the lines of con- 
nections, that the mind claims the right of intellectuality. 
As ages follow ages, the field of the intellects become more 
and more extended, and the province of the mind richer 
and grander. 

Mathematics is one of the most, at least, if not the 
most abstract of the sciences. There the intellect has full 
play. Though abstract, it is not abstruse. Its principles 
are obvious ; its demonstrations natural ; and its conclu- 
sions useful. We see in Mathematics the superiority of 
educated and intellectual faculties. Mathematics give a 
grandeur and sublimity to the mental acquisitions. The 
remotest star is within its reach, and the immensity of the 
universe under its cognizance. 

We lave heard it said, that Geometry is an old 
science, that there can be no improvement to it. When 
a science fulfills all its requirements — then we can say, it 
is complete — it is perfect. Does Geometry fulfill all its 
requirements? Let us see — "Geometry is the science 
which has for its object, the measurement of lines, sur- 
faces, solids, with their various relations." The curve is 
a line — does Geometry give any rule or principle, by which 
we can measure it ? We answer. No ! The circle is a 
surface — does Geometry give any rule or principle, by 
which we can measure it ? We answer, No ! The sphere 
is a solid — does Geometry give any rule or principle, 
which wean measure it ? YVe answer, No ! 



CONCERNI.^G- THE CIRCLE.. 49 



Mathematicians, by tbo means of Analysis— a method 
of reasoning different >rom the ancient Analysis, vvhich 
nifeant an invtBited reasoning ; but now a method of rea- 
soning on geometrical subjects, with the assistance of Al- 
gebra — have given several '' expressions," as the length of 
the curve. Having before remarked on the espression of 
the solutions, we will leave the result, and consider awhile 
tbe means. In the first place, Analysis is diametrically 
opposed to Geometry, The principleSj rules and symbols 
of Algebra, are used 'in Analj-sis ; therefore, the results of 
x\nalysis miK^t be otherwise than ^-eometrical results. They 
are not geometrical conclusions, but they are algebraic ex- 
pressions. Vv^e lose the substance for the shadow, Anj- 
thinfr attractiv^e easilv wins our Kinds. Analysis is a 
beautiful invention. According to LaPlace, ^' The Ge3- 
metrical Synthesis, has the advantage of never losing sight 
of its objocb, and of illuminating the whole path which 
kads from the first axioms to their last consequences; 
vrhereas, the Algebraic Analysis, soon causes us to forget 
the principal object, in order to occupy us with abstract 
combinations. But in thus isolating the objects, after 
having abstracted frooi them what is indispensable to ar- 
rive at the result he is in search of — in abandoning him- 
self to the operations of analysis, and reserving* all his 
forces to overcome the diincuUies which it presents^ the 
analyst is conducted to results inaccessible to synthesis. 
Such is the fecundity of analysis, that it is sufficient to 
translate particular truths into this universal language, in 
order to perceive a series of other new and unexpected 
truths arise from their mere expressions. No language is 
equally susceptible of the elegance, wkich results from the 
development of a long series of expressions intimately 
connected with one another, and all fiowing from the same 
fundamental idea. Analysis, also, unites with these ad- 
vantages, that of being always capable of leading to the 
simplest methods ; for this purpose, it is only required to 
apply it suitably, by a skillful choice of indeterminate 
quantities, and to give the results the form the most conve- 
nient for geometr'cal construction or numerical calculation. 
Modern geometers, convinced of the superiority of analy- 
sis, have especially applied themselves to extend its do- 



•7 



50 * SOME INFORMATION 



main, arid enlarge its limits." To tlie lanj^uage above, we 
have serious objections. It is a species of special 2)lead' 
ing, to win us away from the difficulties we encounter, and 
eonceal from us the imperfections of a system. The dif- 
ference between the geometrical synthesis and the algebraic 
analysis, in a geometrical view ^ is as great as the difference 
between walking by the bright and steady light of the sun, 
and following the deceitful glare of the ignis fatuus. By 
the first, we can traverse the extent of the earth with safe- 
ty — wander over the expanse of the ocean with confidence ; 
while by the other we will be led into pits, mires and 
marshes. 

We admit that the analyst, arrives at results inacces 
sible to synthesis ; for the reason, that in following the al- 
lurements of analysis, he is tempted from his right path, 
and goes from one object to another — attracted by this, 
charmed by that— fancies some accidental beauty : and in- 
tact, pursues such a ^ig-zag, crooked path, that when he 
pauses and considers his where-abouts, the object which he 
started out to find, is not within his sight ; but he finds 
.-something in his bewilderment, which seems to please him 
quite as well, which he adopts instead. This isy»in truth, 
the course of the analyst, when he attempts to discover 
geometrical principles. " Such is the fecundity of analy- 
sis, that it is sufficient to translate particular truths into 
this universal language, in order to perceive a series of 
other new and unexpected truths arise from their mere- ex- 
pressions." Can the principles of Geometry be consistent 
with such a system of reasoning ? Do we arrive at unex- 
pected truths in Geometry ? Do we not know beforehand 
the truths for which we seek ? Are not- our arguments ar- 
ranged to accomplish our object? It may be the case that, 
in pursuing our arguments a truth may be suggested to us; 
but we do not know it to be a truth until it is demonstrat- 
ed according to synthesis. '' Analysis also unites with 
these advantages," &c. &c. We can not for tho life of us, 
see the advantages here alluded to, except they be the 
same as the advantages afforded to the traveller by the ignis 
fatuus, — when all around him is dark, to cheer him with 
its light, to awaken hopes in his bosom, to direct him in 
his path, and lead him into — pits, mires and marshes. 



'■^ 



CONCERNING THE CIRCLE. 51 



''No language is equally susceptible,'* &c. <&c. TMg 
is a beautiful sentence, attractive more from its bappy ex- 
pression, than from a fortunate application. 

'^ Modern Geometers convinced of the superiority,- ' &o 
&c. When we cannot do right^ it is proper to do the best 
we know how. This is the case with '' modern Geometers ;'* 
they do the best they know how. If they are convinced 
of the '' superiority j" we are very obtuse ; for we are not 
The facts of the case are these. Geometers discovered 
the principle of the straight line, by means of which they 
found out its properties, and the relations between it 
and the curve. Not knowing the principle of the curvCv 
they made use of the principle of the straight- liije to 
Snd out the nature of the curve. If the straight- line 
and curve were the same, the principle of the straigbt. line 
would explain the nature of the curve, and the prob- 
lems of the curve would be resolved by the rules of the 
straight line. But we find that the rules of the straisfht 
line will not apply to the curve •therefore, the s-trraight 
line and the curve are not the game Geometerp. kuowio:-i: 
the principle of the straight line and not that of the curve, 
were nonplussed when they came to consider the problems 
of the curve ; yet^ perceiving that the straigbt line and the 
curve were geometrical q^uantities, an<;l that the po^^gon 
and circle were geometrical magilitudgs ; also perceiving 
the intimate relations between them ; they were ooasrrtrir! 
edto believe their identity ; ai1?I cauld not consider theai 
otherwise. When the Algebraic Analysis was invented, 
perceiving that it was not at variance with their pre loy^ 
conceptions of mathematics, and charmed by the wonders 
they acccmplished by its meanSp they hailed it as iihedesid^ 
eratum of the science, ^hey were instantly carried awa). , 
'leartj head and soul,' with tlie invention. Bat/ lo ^ and 
oehold ! after perfecting it with all the knowledge that 
*Vae art an^ science afforded, they could not, even then, 
accomplish their prime object. But to make ail ends meet> 
and make every thing as smooth as possible, they resorted 
to the expediency oi'expressing the solutions : ^i.s if the 
truths of Geometry were mere '^ expressions." ■ 

By Synthesis, we reason ci^ly ; oor arguments are 
iioks of the same rhain : our.conclusiO'B.s are natural ; the 



52 * 80ME INFORMATION 



principles of Geometry are fully established ; our thoughts 
are systematized, and we have in fact a science. By ana« 
lysis, our results have connection, but no relation ; it is a 
combination of expressions, of which each one is complete 
within itself. It partakes more of an art than a science. 
It is an invention. Science is founded upon the discovery 
of certain laws, which are consistent and systematized for 
a certain purpose. An invention is a new combination de- 
pending upon no principle. If we use analysis to enunci- 
ate geometrical principles, we would be like pursuing our 
journey by the glare of the ignis fatuus ; instead of geo- 
metrical conclusions, we would have algebraic expressions. 
With algebraic expressions, we would be as far from our 
object, as we would be in the mire from the end of our 
journey. Geometry is no invention, neither is it depen- 
dent upon invention ; but it i& a science, and dependent 
upon principle. Therefore, synthesis, not analysis, is the 
method of geometrical reasoning. Mathematicians have 
hitherto failed to find any rule or principle to measure the 
curve by the means of synthesis, the same of the circle 
and the sphere. Therefore, the science of Geometry does 
not fulfil all its requirements, it is not complete, it is not 
perfect. Consequently, there can be improvements to 
Geometry. Synthesis is used here in opposition to modern 
analysis, so that modern synthesis embraces ancient analy- 
sis. 

When we apply Algebra to Geometry, whether under 
the form of Analytical Geometry or Calculus, we have be- 
forehand the principles of Geometry to assist us in our 
operations ; we have sufficient relations, connections and 
properties of lines and magnitudes known, and we have 
simply to apply the algebraic formula to these known 
quantities, to engender new formulae These new formu- 
lae proceed only from algebraic princi'ples, and are entirely 
sdependent upon the science of Algebra, and have only 
nominally any connection with Geometry. And to apply 
these algebraic formulae, to discover geometrical proper- 
ties or principles, would be absurd as well as useless. 
Hence arise the difficulties mathetnaticifc.ns expsrience 
when attempting to solve tho problems of the curve line 
\^^ the ADaiy tical Geometry aad Calculus^before they know 



CONCERNING- THE CIROLS 



53 



the prinoiples oi thecurve line. We have no doubt that 
Analytical Seometrj and Calculus will increase the use of 
the principle of the curvej and accomplish ^reat good for 
science. Gur objection to analysis iSj that it was employ- 
ed in illegitimate purposes, when it was used te solve geo- 
metrical problems of the curye line, before the principle of 
the curve was known. 

In conclusiouj we will advert to the science of Astrono- 
my, in reference to- the motions of the planets — the theo- 
ry of circular and elliptical orbits, Kepler's great law of 
the paths of the planets being elliptical would find appli- 
cability a.nd use in the eoincidence, relations and proper- 
ties of the circle and ellipse. It would not be out of place^ 
nor strange to suggest, that the original paths of the plan- 
ets were circular, but owing to the forces of attractions and 
the various perturbations cf the Solar System, these cir.- 
culai' paths were depressed to the elliptical Simply as a 
sugijestion, we offer it to Science. By the properties of 
the circle and ellipse, we may be enabled to clear away a 
great portion of the mist which surrounds the celestial 
bodies: give causes and reasons for^ and perhaps recti- 
fy, the perturbations of their motions ; and remove,^ to 
a great extent, the impediments to the perfect develope- ■ 
ment of the theories in regard to the Laws of the Universe, 




54 SOME INFORMATION 



"A DISSERTATION OW THE PRINCIPLES AND 
SCIENCE OF GEOMETRY." 



-*<'0- 



We propose here, a dissertation on the principles and 
the science of Geometry. And in order that our observ- 
ations may be marked with that candor and fidelitVy with 
that fairness and confidence, which should characterize all 
treatises on science, we will give an extract from a work, 
whicb, from its abilities and researches-, is fully entitled to 
our notice. In presenting this extract, we have adopted 
it as a datum to our further elucidation of the su^bject. 
In Brande^s Encyclopaedia, under article ^^ Geometry,'* 
there is the following : ^^ Objects of Geometry. — In Ge- 
ometry, bodies are considered only in reference to the pro- 
perties of extension or magnitude, figure and divisibility. 
Every body occupies in indefinite space, a certain determ- 
inate place or finite portion of space, which is called its 
volume. The limits or boundaries which distinguish the 
place of a body, and separate it from the surrounding 
space, are called surfaces; a surface is, therefore^ com- 
mon to the two portions of space which it separates. As 
the limitation of space gives rise to the idea of surface, so 
thejimitation of .surface produces lines ; a line being the 
boundary of a surface, or the place in which two surfaces 
intersect each other, and, thereforCj common to both. In 
like manner, the limitation of a line, or the intersection 
of two lines, produces a point. But a point marks only 
position, and has no propertieSc A line has leogth ; a sur- 
face, length and breadth ; and a volume, length, breadth and 
thickness. IleDce^ the properties of lines ; the properties 
of surfaces ; and the properties of volumes or solids, com- 
prehend the objects of Geometry,^' 

^' Although the notion of a point is acquired from the 
consideration of lines — that of a line from the considera- 
tion of surfaces — and that of a surface from the consider- 
ation of bodies or material objects, it does not follow from 
this, that points, lines and surfaces, are themselves really 
material. Geometry regards all bodies in a state of ab- 



CONCEENING THE CIRCLE. 55 



i^ractioD. very diiferent from that in which they actually 
ezist; and the truths which it discovers and demonstrates 
are pure abstractions-— hypothetical truths ; which are not, 
however, on that account, the less useful. For example, 
it is impossible by any mechanical means, to draw a line 
absolutely straight, or to describe a perfect circle ; but the 
nearer the line approaches to perfect straightness, and the 
more accuratelv the circle is described, the nearer will 
their properties approach to those of the ideal straight 
lines and circles, which are the objects of geometrical con- 
sideration. The theorems of Geometry are, therefore, not 
strictly true in their applications to material bodies, but 
they approximate sufficiently to truth, for all practical pur- 
poses. They enable us to ascertain, with all the precision 
of which our senses are capable, the distances of inacces- 
sible objects ; the dimensions of a given surface ; the con- 
tents of a given solid; to compute the distances and mo- 
tions of the planets ; to predict the celestial phenomena ; 
and to navigate a ship fiom any givep. point of the globe 
to any other. ^' 

From the above, we can easily perceive in what light 
we are to consider geometrical subjects. The truths of 
Geometry, by their very extensive use, applicability and 
consistency, demand our earnest compliance. But if we 
go practically in the matter, and require undisputed and 
fully established proofs for Qur demonstrations, we will 
find doubt and uncertainty attending our every investiga- 
tion* 

The province of matter is limited and distinct, and the 
boundaries of materiality are strong and impassible, while 
the domains of the mind are replete with every property 
essential to ail its requirements, 

Let us examine the fundamentals of Geometry — -the 
principles upon which the science is founded. There are 
such things in Geometry as axioms, hypothesises^ and pos- 
tulcUes. They are used in every enunciation of proposi- 
tions and demonstration of problems, and in no case has a 
proposition been established, or a problem solved, without 
either one or the other. And what are these axioms, hy- 
pothesises and postulates, which are so essential to every 



b6 SOME INFORMATION 



proposition and problem ? An axiom is deiined a self- 
evident truth- What are we to understand by this, 
but a something which no demonstration can establish ; 
and containing inherent to itself, those properties which 
presuppose an actual demonstration ? With reason, we 
may say, '* We are anxious for the truth— show us that, 
that thing which you call an axiom, is true.'* We will be 
answered — '' Our minds are so constituted, that we are 
compelled to believe it as true." Then we can justly reply : 
' ^ It is an appeal more to our faith than to our reason.'' 
An hypothesis is a mere supposition, used in the demon- 
stration of a problem, for the establishment of a theory 
— -its truth is made evident,* or its fallacy detected ^ by a 
posterior demonstration. A postulate is an assiimptimi^ 
and takes for granted certain propositions without regard 
to axioms, hypothesises or demonstrations. Geometry as 
a science, is dependent upon them. If we use in geometry 
such truths only, which have been demonstrated, y/e will 
have no truth whatever ; Geometry will have no existence ; 
those important facts which shed a halo of glory oyer the 
/ genius of man, w^ill vanish ; the knowledge of mankind in 

regard to the motions, orbits, distances and forms of the 
planets, will be lost ; and, in fact, all the wisdom and 
erudition of the world will be jeopardized. Because, 
every science, and -all knowledge, are founded upon axioms. 
Such being the bases of the sciences, their superstructures 
must be of corresponding natures; their materials must 
partake of the elements of the foundations ; otherwise, 
there will be no consistency nor unity; their parts will be 
Incongruous and distinct; and their masses chaotic. Not 
that we would be understood to mean, that the sciences 
are assumptions and suppositions, but that in our admira- 
tion of their consistency and beauty — of the marvelous 
"wonders they are able to achieve — of the usVfulni^ss and 
necessity they are to our existence — of the simplicity and 
ease by which they unfold the deepest secrets of nature — 
of the amelioration of sufferings and the diffusion of hap- 
piness which they accomplish, and of the connecting link 
which they form between the great creative Genius of the 
aniyerse and the creations of the earth, we must not for- 
get that their foundations are liable to scepticism and die- 



CONCERNmO THE GIRCLE. ' n7 



cnssioDj and that their principles are established by the 
eoDsistece}^ of the conclusions.. We have indulged these 
few remarks, not with the intent to attack the grand troths 
of science, but to show that we are not to expect an a 
priori demonstrsition for every truth of Geometry, Evi- 
dence and coDsistency of conclusions establish a truth as 
iirm as the most direct demonstration of which reason is 
capable ; for, in fact, the most direct demonstration of Ge- 
ometry, is based upon evidence and consistency of conclu- 
sions. A geometrical demonstration is considered the 
most conclusive and most satisfactory argument of truth, 
of which the reason of man is capable ; and even that, v/hen 
passed through the Jiery ordeal, will exhibit points of par- 
tial destruction, and be disfigured with chars. 

Takej for instance, any proposition of Geometry, its 
, premiss is either an axiom, some fcruth which has been be- 
fore demonstrated, or an hypothesis which is yet to be de- 
monstrated. If it is an axiom, we must take it upon evi- 
ckiice ; if it is a truth before demonstrated, its demonstra- 
tion is based upon an axiom or hypothesis ; therefore, we 
must take it- for s:ranted, as evidence alone will entitle it 
to our belief; or if it is an hypothesisj it is a supposition 
-—so much for the premiss. The arguments for the proof 
of the proposition, are natural conseqences of the premiss, 
and are based upon assumption or supposition, as the case 
mav be; some times the natural consequences of the pre- 
miss will lead to an axiom^ or a demonstrated truth ; if to 
an axiom, it will be taken for granted without demonstra- 
tion ; if to a demonstrated truth, its demonstration is resolv- 
able to an axiom, hypothesis or postulate. And the con- 
clusion will be in accordance with the ar/^uments of the de- 
monstration, and like them, will be based upon assumption 
or supposition. Hence, if we strictly analyze the demon- 
stiai^ion of a geometrical proposition, we will have no rea- 
son to demand an absolute demonstration for every truth 
of which our intellect is conscious. But, in order that 
truth may be acceptable, and that a collection of facts may 
be systemized ; or that certain observations of facts, mar 
be reduced to such an order as to form a science, there is 
a certain number of terniB, called definitions, explaining 
such things which are abBolutely neeessary for a correct 



58 • SOME INFORMATION 



understandiDg of the subject we may have under eonsid 
oration. For instance, if we open a book on Geometryj 
we will see certain marks, some straight, some curved, and 
some, as it were, bent or broken. Now, it is necessary for 
the geometer to explain why lie uses certain kinds of marks 
for some purpose, and other kinds for some other purpose ; 
to give names to them ; to describe certain properties which 
they have in common, and others which they, have in dis- 
tinction When this is done, we are prepared to under- 
stand him. When he alludes to a straight line, we know 
the difference between it and a curve; when he uses a 
triangle, we at once perceive that it has properties very 
unlike those of a square or circle. And in this manner, 
we are led to understand, that a straight line and curve 
line have certain relations and uses, which are important 
to be known. And in order that these relations and uses 
may be satisfactorily interpreted, there are certain terms, 
expressive of certain facts or states of knowledge, by means 
of which, the mind intuitively perceives a connection be- 
tween the things known and those for further elucidation ; 
siich as axioms, hypothesises and postulates ; as demon- 
strations, theorems, problems and lemmas; as corollaries 
and scholiums. With the assistance oi these, the mind is 
carried step by step, in all its investigation of extension ; 
and is able to discover by such investigation, the proper- 
ties and relations of geometrical magnitudes. They are 
the data by which the hidden truths are revealed. Upon 
them a system of logic or argumentation is conducted, and 
by the conformity of the arguments and conclusions with 
these accepted truths, we have the scicDce of Geometry. 
After informing us of the properties, relations and uses of 
the lines or "jnagnitudes of one dimennon^ the geometer 
then introduces to us the knowledge of the properties, re- 
lations and uses, of suo'faces^ or nwgnihides of two dimen- 
sions ; and after that, to the knowledge of the properties, 
relations and uses of solids^ or magnitudes of three dimen- 
sions. This course would embrace the perfection of the 
system, or the entire objects of Geometry. And could 
this arrangement be carried successfully in operation, the 
science of Geometry would be complete, Although Ge- 
ometry can boast high antiquity, and though many of its 



CM>NCSRNmO THE CIRCLEc 59 



fields have been explored, there is yet much in store for 
the industrious and persevering adventurer, over the fertile 
domains of this grand and noble science. 

Geometrjj so far as it is adopted by the schools and 
colleges of the present day, is confined to the properties, 
relations and uses of the magnitudes dependent on the 
straiorht line : and to the relations and uses of the magni- 
tudes dependent upon the curve ; and considers the proper- 
ties of the curve similar to the properties of the straight 
line. Here is the imperfection of the present geometrical 
system ; because, from the very definitions of the straight 
line and the curve, we see there is a difference between 
them; and having a difference, they cannot have similar 
properties; and not having similar properties, the treat- 
ment of the problems concerning the curve, must be very 
different from the treatment of the problems concerning 
the straight line. To show that we are very anxious for 
the reputation of truth, fairness and candor, we will give 
another extract from the same authority we used before. 
Under '* Metliod-'^ of Demonstration ^^^ in the same article- 
Geometry —there is as follows : " In their more difficult 
researches, and particularly in those relative to curve lines 
and surfaces, the ancient geometers had recourse to the 
method cf exhaustions. Admitting no demonstrations but 
such as are perfectly rigorous, they did not consider it con- 
sistent with the strictness of geometrical reasoning, to re- 
gard curve lines as polygons of a verj great number of 
sides; but when they proposed to investigate the proper- 
ties of any curve, they regarded it as the fixed term to 
which the inscribed and circumscribed polygons continu- 
ally approach, in increasing the number of their sides. 
The continual approximation of these polygons to the 
curve, afforded an idea of the properties of the latter^ the 
more accurate as the number of sides was greater. But 
it still remained to prove, by some recognized principle of 
demonstration, the truth of the properties that had thus in 
a manner been divined ; and this was done by showing that 
every supposition contrary to them, necessarily led to a 
contradiction. In this manner, they demonstrated that 
the areas of different circles are to each other as the 
aquai^s of their respective diameters, the volumes of 



'7 



60 SOME mFORMATION 



spheres as the cubes of their diameters , that pyramids of 
the same height are as their bases, &c. (OAr^NOT, liefiex- 
ions sur la Metaphysique du Calcul Infifzitesinic.L''') 

Now, it is seen, that the ancient Geometers did not 
consider their demonstratior! of the curve as rigorously 
true ; and their ideas of the properties of the curve, were 
formed merely from a semblance of the truth — from the 
approximation of the polygons to the circle. They adopted 
this as the property of the curve, simply from -the facty 
that they were not able, hj any reasoning they had at 
command, to distinguish between the infinitesimal quanti- 
ties of the straight line ond the curve. They reduced the 
line so small unUl thev lost sio-hc of it altogether. We 
have shown under " A Discussion of the Properties or the 
Straight Line and the Curve," how the properties of the 
curve and the straight line may be confounded, from the 
intimate relationship which exists . between therii. And, 
although this intimate relationship exist&\ it does not argue 
that they are the same. The properties of the curve are 
uniform and consistent. The ancient Geometers consid- 
ered the effects of the properties, not the properties them- 
selves, nor even the causes Hence, ressonin^: from these 
effects, when they found all suppositions contrary to their 
hypothesis, leading to contradictions, they immediately 
concluded that the properties of the curve and straight 
lines were the same; when, in fact, the properties of the 
curve were not under discussion, but the effects of these 
properties. Though the effects of the properties may have 
enabled them to ascertain some truths in regard to the 
curve, yet these truths will bo limited, from the fact, that 
all the truths of the curve are dependent upon the prop- 
erties of the curve ; and until these properties- are known, 
no reliable system can be deduced, by means of which all 
the problems concerning the curve, can be resolved. The 
ancient geometers, from these effects^ and the application 
of the rules and the principles of the straight line, de- 
duced a system of principles and rules, altogether agree- 
able to their previous conceptions of Geometry, and con- 
sidered ail problems of the curve, the same in nature as 
the problems of the straight line. The modern geometcrSy 
iTMeiving all their knowledge of the science from the 



CONCEaNlNG THE CIROLE. 61 



ancient, became reconciled to their supposition^ concerning 
the properties of the curve, but found that even when the 
supposition was admitted, they could not solve the prob- 
lems of the circle ; and they in turn had recourse to the 
Algeb'aic Analysis. And lite the ancients, became per- 
fectly enraptured with their arguments. We have ezam= 
ined the Algebraic Analysis, under- — " A Critical Exam- 
ination of the Algebraic Analysis ^^^ and have given our 
reasons why that method of demonstration is not a leo-iti= 
mate one, when used for the discovery of geometrical 
principles; and. even that demonstration is based upon the 
supposition, that the properties of the curve are similar to 
the properties of the straight line. The truth is, theAji° 
cient geometers with all their ingenuity and learning, could 
not discover the properties of the curve ; and the modem 
geometers, with all their researches and knowledge, were 
hot able to go any further than the ancient. 

The principle of Geometry, that the arc of a circle is 
always greater than its chord, is well known, and that how- 
ever infinitely small \kiQ. chord may be reduced, it will al- 
ways be so. But the properties of the curve were not 
tnown ; and in the absence of this knowledge, the geome- 
ters, both ancient and modern, controlled their arguments 
entirely by what was known of the properties of the 
straight line. No geometer of the present day, does con« 
sider the circle otherwise than a polygon of an infinite 
number of sides, although there is no proof iov so doing ; 
but the geometers are reconciled to it, because the ancients 
were satisfied. Our ai,m is to show the errors of the present 
system, and introduce the properties of the curve into ge- 
ometrical consideration. For Geometry is established 
upon truth, and the aim of the geometer ski}uld be to re- 
cognize truth. 

We have in Geometry, squares formed by straio-ht 
lines, or magDitudes^. of on^ dimension; we have in Arith- 
metic, squares formed by numbers as well as by iinit^^ but 
we have not in Geometry squares formed by surfaces or 
magnitudes of two dimetisions. The latter are properties 
which are fully entitled to minute investigation. When 
we undertakii to introduce a new principle into an estab- 
lished gcie»ce, it is quite necessary and proper that wo give 



62 SOME lNF0RMATiO^ 



.f. .^. 



reasons for our introduction, and show its peculiar impor 
tance. Every principle has its legitimate use ; and its very 
existence argues a natural application. A principle lavs 
the foundation for many and important discoveries, and 
when fully digested, the knowledge of man is enriched bv 
truths which would perhaps have remained for ever in the 
ore^ had it not; been interpreted and applied 

Geometry is that branch of science which considers 
magnitudes in their extensions,|measurements, relative po- 
sitions, propertiesi^and connections. Arithmetic.is another 
branch of science, which has for its object, the consider:), 
tion of units and numbers, their relations and proDerties 
which tend to point out their ise, practicability and nece-/ 
^ to common life. The utility of Arithmetic is fullv 
displayed in the importance and grandeur of its results 
in the feasibility, beauty and perfection of its operations' 
and in the conciseness and brevity of its deductions' 
When we consider the near relation 'and connection exist- 
ing between Geometry and Arithmetic, we are apt to con- 
found the one with the other/ For instance, when we 
speak of a square in Geometry, our mind is immediately 
earned to an arithmetical square, and our idea formed of 
one resulting from units or numbers— in other words, oifr 
mind takes cognizance m^re readily of the contents or'area 
of a magnitude, than its dimensions*. Now, this is easily 
accounted for ; from our infancy we are taught to compre^ 
faend the idea of units and numbers, we have their uses 
relations and properties, presented to us under tlie most 
favorable circumstances — that is, when our mind is voun^ 
impressible and tractable. They, at that time, take a 
strong hold of our thoughts. The technical terms of 
Arithmetic become familiar to us; when ever they are • 
usedj our idea is immediately formed of their imports 
•from the considerations we formed of them. NovT, in Ge- 
ometry and Arithmetic, there are several terms or expres- 
sions of operations, which, in some ^noinfs or particulars, 
are common or applicable to either science; yet, in their 
results, owing to the principles of the different branches of 
science, are not corresponding. Thus, in addition, we have 
^^j arithmetical rules, the [number six (6) added to the 
number two (2), producing the number eight (8). 



r' 



m 



CONCERNING THE CIRCLE. 



63 



In Geometry, the magnitude 
Fig. 3 




added to the magnitude 



Fig. 4. 



•^r:f' « ^Ht 



tC^' 



3 



Fig. 5, 



producing the magnitude 




Now, 



if the principles of the two branches of science were the 
same, we would have six (6) added to two (2), producing 
(32^ Hence we perceive the principles or modus operandi 
of the two branches of science ; or again — 



6 -i- 3 -U 9 -!- 7 = 25 = 8 -h ^ -t- 4 H- 6 -= 25 



and 



Fig. 6 




+(3+0= 




T/£D^; 




The difference between an arithmetical addition and a 
geometrical addition. 

The square of triangles^ and other puzzles of a like 
nature, are entitled to geometrical consideration ; for thej 
establish the principle of squares formed by surfaces or 
magnitudes of two dimensions. 

In Arithmetic, we have by addition, an increase of 
units ; in G-eometry, an increase of magnitudes or dimen- 
sions. G-eometry is the science of lines ^ in their connec- 
tions with, and relations to, magnitudes. Arithmetic is the 
science of units^ in their connections withj and rela 
tions to, numbers. G-eometry treats of magnitudes by 
themselves considered; Arithmetic treats of the iiumber 
of the component units of those magnitudes^ I/inesbear 
a corresponding relation to magnitudes , that units bear to 
numbers; in other words, the increase of the lines enlarges 
tlbie magnitudes, as the addition of units increase the TDurn- 



64: 



S0M:E INFOilJVr.ATIO.iN 



berSy and vice versa. Moreover, lines in Geometry are 
the units of Arithmetic , and the units in Arithmetic are 
the lines of Geometry 

We have in Arithmetic^ different orders of number»3> 
as we have in Geometry different orders of magnitudes ; 
in the former, they are designated by the terms, units, tens, 
hundreds, thousands, and so on; in the latter, by the 
terms, lines, surfaces and solids. In Arithmetic, each suc- 
ceeding term is a higher and uniform grade of the pre- 
ceding ; in Geometry, the lines compose the surfaces, and 
the surfaces compose the solids ; among the lines^ thr$e 
straight lines compose the simplest surface- — the triangle ; 
while one curve line composes the simplest surface — the cir- 
cle. Then we have other kinds of surfaces and solids, some 
composed by straight lines entirely, as the square, pentagon^ 
hexagon, &c.; and some composed by straight lines and 
curves, as the segment, sector, &e,, among the surfaces; 
and the pyramid, cube, parallelopipedon and prism, formed 
by rectilinear surfaces ; and the sphere, formed by a curvi- 
linear surface ; and the cone, cylinder and hemisphere > 
formed by rectilinear and curvilinear surfaces, among tho 
solids* 

Let us retrace our steps. We were considering a get- 
metrical square, or a square formed by lines, in distinctioia 
from an arithmetical square, or a square formed by units. 
Take the latter, we have — 

(6)^ = 36 or (2 H- 4)- •-= 36 or (3 H- 3)^ == 36. 

This has reference to the number of units, which ie the 
province of Arithm.etic to d etermine 



A—Fig. 7 



B--Fig 8. 



: 








i 

1 



i 






1 










■ 







In Geometry, we have the magnitudes A B — suppos- 
ing them, to contain the same number of unit, of meamrc, 



CONCERNING THE CIRCLE, 



65 



Then if we form squares by them, each separately^ we will 
have the square a b c d, formed by A, and the square 
e f g h, formed by Be 



Fig. 9. 



a 



c 



b 



e 





! ' 




1 












I i 




1 1 





Figc 10. 





1" 


1 


1 




1 


ft 


1 


r 


1 


\ 


1 


1 




1 


1 



h 



When they are placed to form 6 units on a side^ they 
will form equivalent squares^ having in a b c d, 4 times the 
magnitude A, with one unit over, making 25 units ; and 
in e f g h, 3 times the magnitude B, with seven units over^ 
equal to abed. The seven units over in e f g h, are 
equivalent to the magnitude B plus one unit of measure, 
making 25 ; or let them be placed thus- — forming siz units 
on a side. 



A— Fig. 11 



B— Fig. 12. 



1 


1 


i 

1 




Mil 


1 1 ) 



1 


1 


1. 


J 1 ( 


! 


J. 

« 




( 1 
: [ 


1 


f. 

1' 


- ^ C 


i 1 


. 


1 




1 ' 



Here we have squares formed equivalent to 36 units ; 
the number of units over in B is equal to 12 — an equiva- 
lent of two B, since A is used sis times in forming figure 
11, while B only four times in forming figure 12- — the 12 
units making six times. Try A aud B again. 



#6 



SOME INFORMATION 



Fig. 13. 



1 

: ^ 






I 1 










16 ;' 



: ■:4' 



Fig. 14. 



I I 



We have by the above, the squares of Fi^urp n 
(16-1-4-1-4-1-1 -1-1) = 26. In Fig. 14-(9ll-9 — 
4 -I- 4) = 26, equivalent squares again. It may be asked 
why not erect a square on the two units to the right of 
the first line in Figure 13. I will answer, they being single 
units, and already had advantage in the square of 16 let 
fall. Try other naagnitudes. 



C— Fig. 15. D— Fig. 16. 






^^ 





Squares on C and D as follows • 

On C, (25-1- 4-i- 9 -|- i -|_ i) _-4o^ on D, 
(16 «!- 16 -1-4-1- 4) == 40. Thus squares as above, can 
be formed from magnitudes and D. Try magnitudes 
E and F. . "^ "^ 



E— Fig. t7. 



•Fig. 18, 



J1J_ 



jjliT4d~rT 











1 


1 1 1 


1 


i 


i -11 


'. ■ ^ — 


. 4^1*^1 t 



^; ■; Bet^splace'^Eso as to torm/iiquare with 'thirteen 
^ja,Sa^'«ide— thus ; * ""^ 






/! 



CONCEJiNIHG THE CJ?X'LE« 



67 



Fig. 19, 







Then -we will have a space of twenty-five . squares id tbe 

oeixter. Now, if we take F and plan- it thus. 

Fig. 20^ ' /%'/ . 




'/ 



68 



SOME INFORMATION 



SO as to form a square of twelve squares on a yide. If 
we examine the magnitudes thus formed, we will perceive 
that each is composed by E and F, taken an equal num- 
ber of times ; and that the number of squares comprised 
in the magnitude formed by E, is equal to the number of 
squares in the magnitude formed by F. Because, although 
the square formed by E has thirteen squares on a side, 
and the square formed by F has only twelve, each square 
18 equal ; for the thirteenth squares on the side of the 
square formed by E, being put in the center of the square, 
will exactly fill up the space in the center, leaving twelve 
squares on a side, and equal to the square formed by F. 

Other experiments like the foregoing can be tried, 
which go to prove that the principles of Geometry are 
consistent; and that equal magnitudes will form equal 
squares ; a truth corresponding to the truth, that equal 
lines will form equal squares ; and to the truth, that equal 
numbers will form equal squares. 

We will now turn our attention to squares formed by 
curvilinear magnitudes, or magnitudes having curved sides. 
Take for illustration the followino; mao^nitudes : 



Fig. 21. 



Fig. 22. 





Let us examine A and B, and we will find that the 
circumscribed squares are equal; and that any circum- 
scribed squares of A or B will be equal — the same of any 
inscribed square. Now, knowing that A and B are equal, 
we must naturally conclude that equal circular magnitudes 
will form equal squares. And comparing A ©r B with G 



CONCERNING THE cmCLE, 



69 



< 



%. 23. 



Fig, 24, 





or D; or C and D together ^ we find tbat none of their 
squares correspond ; or, in other words, their squares are 
not equal. And knowing that the magnitudes are not 
equal, we must also conclude, that magnitudes not form- 
ing equal squares are not equal. 

These experiments have reference to ^^ formation of 
magnitudes, which is the province of Geometry to de- 
scribe; while experiments of numbers have reference to 
the quantity of units, which is the province of Arithme- 
tic to determine. These experiments simply prove, that 
magnitudes forming equal geometrical squares are equaL 
In Arithmetic the reason why the results are the same, or 
that the same squares are obtained is, that in forming an 
arithmetical square, we have always the units to deal 
with \ consequently the number of those units are un- 
changed, because the area or component parts are alv)ays 
brought in consideration. While in Geometry, we have 
the lines or formation of the magnitudes, which are not 
affected by the dimensions of the magnitudes— their in- 
crease or diminution. We now come to the ^' Quadrature 
of the Circle, [See next page.] 



;^ 



70 



SOME m'FOlimATiO'^ 



I'm: eWADRATOBE OF THE CIBCLE. 



Fisf. 25 




In the sq-aares B F G H, and A B C D, we find E F 
G H is doable A B C D, haying twice the numher of equal 
triangles; and A B C D, for like reasoBj double A M F, 
the square of the radius of the circle — /. we get this pro- 
portion : 

EFGH:ABCD::ABCI);AMCP, 
or square of diameter :ABCI) ::ABCD, square 
of radius .\ A B C D, is the mean-proportional of the 
square of diameter and the square of radius. 



CONOaRKlNa THE CIRCLE. 



71 



Now, in the diagram before na, (Fig. 25,) we perceive that 
the circle is composed of the square ABC D, plus the four 
segments of the circlOo Now, knowing the contents of 
the segmeatSj we can very easy be able to find the area of 
the circlco Novf, in the diagram below, we perceive that 

Fig. 26. ■ ^ . 



A' 



B 




X^ 



the triangles B B A, A F 0, B a D and P H, Fig. 25, are 
inverted towards the center M^ from which we find the cor 
respondence of the sq^uares formed by the segments and 
the magnitudes E A B, &c. We perceive that their cir- 
cumscribed and inscribed sq^uares are equal And if we 
place them in their first position, we also find the squares 
which they form equal. From principle before estab 
lished, we can conclude that the segments are each equal 



72 



SOME INFORMATION 



to the magnitudeB E A B, &o. But before we decide 
positively, let us examine the* magnitudes more olosely. 
Take the segments as follows : 



Fig 27. 



< 




I^ we in this diagrain, make the triangles B X A, 
ARC, C T D and DUB, in the same segments, we 
will find by comparing the squares which they form, that 
their squares U T R X are equal ; that their squares A 
BCD are equal, but that their squares L Tj P and 
I K W Z are not equal If we examine these squares^ 
we will perceive why some are equal and others not. The 
segments and triangles have equal altitudes and equal bases^ 
which accounts for the equality of the squares U T R X and 
of the square A B C D; while they have not the same or 
equal magnitudes-; which exhibits clearly the reason of the 
difference of tlie squares LOOP and I E W Z If we 



UONCERNING- THE CIE(5LE. 73 



-oompare the triangles B X A, &'C,j with the magnitudes 
E B O Aj &a, in Diagram No. 25, we will find that tlie 
correspondence of their squares will be the same as that 
just shown ; which go to prore clearly that the magnitudes 
B B A and BOA, Diagram No>25, are equal. The near- 
er we make the equality of the triangles and the segments., 
the nearer will become the equality of their squares. 

Now, the square A B C D plus the segments of the 
circle, is equal to the circle. And the triangle A E Bj, is 
equal to one half of the square of radius : therefore, each 
segment is equal to one-fourth of the square of radius ] 
or the four segments equal to the square of radius, From 
which we can deduce the following geometrical rule : 

Find the mean-proportional of the diameter and ra- 
dius : and to that mean proportional add one-half of the 
difference of the diameter and that mean-proportional 
The square formed by that line will be equal to the given 
oireie. 

For the convenience of Arithmetic, we can deduce 
the following rules — either will solve the problem : 

Square the mean-proportional of the diameter and 
radiuSj and add square of radius ; or multiply square of 
radius by three ^ or square the diameter , and subtract the 
square of the radius : or three-fourth square of diameter. 

We will here introduce another evidence to show the 
equality of the segments A B, &c.; and the figures 
A B E, &c.. in Diagram No. 25. We introduce it here^ 
only for the reason ^ that the more arguments we can pro- 
duce concentrating to one and the same point, the better 
will that point be established, and the more conTinoing 
will be the proof of its truth. We have already proven 
in the first part of this pamphlet, the equality of the 
segments A B, &c,j and the figures A B E, &q., 
according to the principles of Geometry^ and with all 
the' , ^trictness of a rigid geometrical demonstration . 
Our/ aim w to do all we can for the advancement of 
the great science of Geometry ; and bring forth any 
new principle that can reveal its truths^ and add to its im- 
portance and usoo 'WJiQn 'we place two quadraxits at the 
distance of the raiiius apart, and draw li.rt.es" as by Fig. 28, 
[See n.ezt page<] 



74 



SOME INFOB^MATiON 




I 



|i 



we will find that the magnitudes which they form, can be 
reduced as follows ; AGBDHG =--ALB.DMO-= 
A E B 0, the square of the radius. And when we place 
two figures corresponding to the figures A E B, (&c., in 
Diagram No. 25 — at the same distance apart, and draw 
lines as by diagram above, we will find that the ma^itudes 
which they form, can be reduced as follows : 

AaBDHC = ALBDMC = AEBC^^/^^ 
square of the radius. From which^ it may be argued with 
much justness and great reason, that two segments 
A O B, &c,, forming a square equal to a square formed by 
two of the figures A B E, &c., are necessarily and eyi 
dently equal to two of the figures A O B E, &c. 

Three different ways substantiating one point , a most 
convincmg proof? and establishes that point beyond all 
cavil and dispute. It is true that these demonstrations 
are new to Geometry ; but because they jare new, that is 
no reason why they are untrue. If the problem could 
have been solved by any method of demonstration known 
to the s-oienee of Geometry, most uadoubtedlj^ the prob 
lem of the quadrature of the circle would have been solved 
long ago. But, for the reason thatj the great mathemati 
cians confined them^selves to the known rules, the solution 
of the problem remained una-ccomplished ; and their grestt 
mbadi? u.nsuccessfo.l to dera.onstrate the' possibility of 



f30NCBRNINO THE CIRCLE. 



75 



its solution. The circle is finite, it has a definite area; 
and why should it not be determined as well as the area of 
any square or triangle ? No geometer has demonstrated 
its impossibility. VVe will admit that no received method 
of demonstration can solve the problem ; and that fact is 
an argument, it ia more, it is an evidence in our favor. 

We have presented new methods of demonstration, 
and they solve the problem. What more can Science ask ? 
What more can Truth require ? 



76 SOME INFORMATION 



GONCERMING THE ELLIPSES 

0- - 

On page -^\j <ji tuis pampniecj i uiiaertia upon tiie 
Ellipse, wlios3 eccentricity is equal to oaehalf. In that 
'^ 33 the conjugate axis is the diagonal of the squ<ire for m^ 
•.■ji by one half the transverse axis; and the conjugate axis 
is equal to the distance between the foci. "Whenever the 
eccentricity is greater than one-half, the rectilinear figure 
formed bv one-half the transverse axis, connectinsj the foci 
with the extremities of the ccnjugate axis, will be a rhom- 
bus instead of a square. The rccentricity is the ratio be- 
tween the distance of the center and the focus of the 
ellipse and the semi-axis major ; and the properties of the 
ellipses and rhombuses thus formed, are different from the 
properties of the ellipse and square treated of on page 20. 
The ellipse of any eccentricity, is the circle with the trans- 
verse diameter extended, and the conjugate diameter short- 
ened — in other words, the ellipse is a depressed circle. 
And in the same proportion we shorten the conjugate di- 
ameter — we lengthen the transverse diameter. The ellipse 
of page 20 is what we may call the geometrical ellipse \ 
or the ellipse having its axes maintaining a geometrical re- 
lation ; while the ellipses treated of here, we may call 
o.rithiwticol ellipses : or ellipses having their axes main 
taining an arithmetical relation. Because, with the form- 
er, the difference between the axes is a £]:eometrical ratio ; 
with the latter, the difference betweersthe axes is an arith- 
metical difference. Suppose we have a circle of 12,12 -|- 
inches diameter — the ellipse of page 20 will have for its 
transverse axis 14, and conjugate axis 9.89 -|-. . (See 
page 25.) Here we have 2. geGraetrical ratio between tl^e 
transverse and conjugate axes. T^rat is, the conjugate 
axis is the mean-proportional of the transverse axm and 
one half the transverse axis. With the same circle, we 
can form ellipses having axes as fellows: 13.12 -|- and 
i:.lv_|_; 14.12-]- and 10. 12 -|-; 15.12 -j- and 9. 12-;-; 
16.12 -;- and 8.12 -|-, &c., &c. Here we have an arith- 
vieti'^al diiference between the transverse and conjugate 



CONCERNING THE CIRCLE* Hi 



axes. Thafe is, the axis minor or conjugate axis is formed 
^j syhtt'wctinx a defioite quantity from the diameter ; and 
ft he axis major or transverse axis is formed by adding tha 
•same quantity to the diameter. And when we know the 
two axes, and wish to find .the diameter of the circle hav- 
ing a periphery equal to the periphery of the ellipse, we 
will add them together; and one-half the sum of the axes 
will be the diameter of the circle. And with this diameter 
ana the ratio 3.4641 -\-^of the circuwference an^^ diam- 
eter^ ue can find the length of the periphery of the circle 
equal to the periphery of the ellipse. 

The area of any ellipse, is the mean-proportional of 
th3 areas of the circles constructed upon the axes ; in 
other words, the area of any ellipse is equal to square 
root of three times square of one-half transverse oxis^ niul- 
tip'ied by three times square of one-half conjugate axis, 
((See page 24 ) 

The ellipse is one of the conic sections; and is, as its 
name signifies, a deviation from the circle. It has engag- 
ed much of the attention of the ancieut Greek geometers; 
and many properties have been discovered, owing to the 
peculiar formation of its periphery. These properties^ 
thus discovered, are not the properties of the curve form- 
ing the ellipse ; but are the properties of certain straight 
lines drawn in the ellipse, and terminating at the curve. 
The peculiar formation of the curve, has given these 
straight lines many properties which have been considered 
by Analytical Geometry and the Calculus. But the pro- 
perties of these straight lines have been confounded with 
the curve line. And the nature and properties of the 
ellipse have been attempted to be interpreted through the 
means of the properties of these straight lines. The same 
error which has possessed the minds of the ancient and 
modern geometers, in regard to the curve line which forms 
the circle, acts with similar force in their investigation of 
the curve line which forms the ellipse. They having en- 
grafted in their minds a similarity of the straight line 
and the curve ; this idea has attended them in all their ex- 
aminations of the respective properties, until they are 
loth to aokaowledge the inoorrectness of their coacla- 
sions. Under article " Conic Sections/' in Brande's Bn- 



78 SOME INFORMATION 



cyclopaedia, the Parabola is first considered; after form- 
ing the parabola, certain straight liiies are drawn in con- 
nection with- the figure, and the eooclusion obtained is, 
** that in the parabola ^ the square of anif ordinnte L G «5 
equal to the rectangle qf the cor?esponding absciss E Gr in- 
to a constant quantity. From this all the other properties 
of the curve may be deduced. It is in fact, the common 
equation of the parabola.". (See Parabola.) Now> in the 
first place, all the lines used in the demonstration of the 
parabolrv, are straight lines ; and whatever truth may be 
eliandatedby these straight liws, thattriitli must evidently 
be a truth oj those stroAght lines. Yet, in defiance of rea- 
son and science, it is asserted that the properties thus dis- 
covered, are properties of the curve. And in the second 
place, these properties may be called the properties of the 
parabola ; but they cannot be considered the properties of 
the curve of the parabola. For not once in the demon- 
stration of the parabola, is a single property of the curve 
employed. They are truths of the straight lines, and can- 
not be considered otherwise. The algebraic equation of the 
parabola, is deduced from the property that " the square 
of th^ ORDINATE is equol to the rectangle under the absciss 
and the parameter." The polar equation of the parabola 
is derived from the system of rectangular co-ordinates. 
(See Davies' Analytical Geometry.) Here are nothing but 
straight lines again ; and the polar equation is, therefore, 
an equation of straight lines, not of the eur^e. 

The parabolic curve has been rectified, although it had 
been considered impossible. It was accomplished by an 
Englishman, named William Neil, who died in 1670, at 
the early age of thirty-three. The quadrature of the para- 
bola was accomplished by Archimet)ES. Other curves 
have been rectified, and they were all considered impossi- 
ble until they were accomplished. Which is quite natural. 

The rectification of the cycloid was accomplished by 
Sir Christopher Wren ; and a great number of other curves 
have been rectified. The exact quadrature of the lune has 
been found. And that the rectification and quadrature of 
the circle are impossibilities, there is no evidence in na- 



CONCERNING THE CIRCLE, 79 

4> 



ture, reason or Bcieoee, to support such an idea; cor no 
reliable testimony ia the writings of mathematicians to 
give credence to the notion. 

The properties of the ellipse and hyperbola are de^; 
duced from similar straight lines as tho^e of the parabola. 
And the algebraic equations of the ellipse, hyperbola and. 
circle, are derived froei the properties of certain. straight 
lioes. The ellipse from the property ^^ that the rectangle 
under A H and R a is to the square. of the ordinate H D 
in the ratio of the s(|uare of C A to- the square of jO B'' 
The hyperbola from the property ,'^ that the rectangle un- 
der A H and H. A is to the ordinate H D, in the ratia^of 
the square of C A is to the square of C B»'V AndXhevCir- 
cle is derivtd frosi the .co-ordinates of tliO ppints^ of thj 
curve. Tho polar equations of the ellipse, hyperbola, an4 
the circle, are all derived from the system of rectangular 
co-ordinates. (See Davies' Analytical Geometry.) Or 



those of the ellipse and hyperbola are deSned- from the 
eq-aation between the Q^adius vector-— ihd^i is^ Si'line drawn 
from the focus to the curvoj and the angle whi/jh it makes 
with the transverse axis, (See Branda's Encyclopasdia.) 
Here we have in all these properties tbe properties of 
straight lines only,, and in none do perceive any 

of the properties of the curve; but we perceive rela- 
tions between the straight line and the curve ; and only 
those relations and the properties of the straight, line^ 
are used in the elucidation of the subjects of Geome- 
try. The properties of the curve are in no wise consider- 
ed in the present state of the science ; which is so unrea- 
sonable and unsatisfactory for the perfect development of 
a. system of scientific facts. The properties of the curve 
have high claims to be considered and admitted in the do- 
main of the science. Without them^ Geometry is imper- 
fect, and the grade which it occupies in the rank of sciences, 
is consequently impaired. It becomes a secondary science, 
when the assistance of other sciences are required to elu- 
cidate truths which are exclusively and entirely its own. 

In regard to the lune or lunula of Hippocrates of 
Chios, we will remark in passing: This figure is in form 
of a crescent, bounded by two arcs intersecting at its ex- 
tremities. It is celebrated in the history of Mathematics, 



80 SOME INFORMATION 



on account of a remarkable property discovered by Hip- 
pocRATEs, that it can be exactly squared, though the quad- 
rature of the whole circle has baffled the ingenuity of 
mathematiciant- of all ages. (See Brande's Kncyelopsedia.) 
Now, when the area of a ngure like the lune can be accu- 
rately found, formed as it is by two arcs of circles; how 
is it, and why is it, that the circle, formed by a curve simi- 
lar to the arcs which form the lune, is impossible to be 
squared ? "We can state why the circle was impossible to 
be squared. Because mathematicians resorted to every 
expedience but the right way. If Hippocrates had tried 
any other way but the way he did, the squaring of the 
lune, in all probability, would have been among the strw^ge 
chimeras of some distempered brain^ or some absurdity fes- 
tering the imagination of the mind. But fortunately for 
the zealots of truth and lovers of science, Hippocrates 
luckily discovered the true and only way to solve the prob- 
lem. If it was required to find the dimensions of space^ 
then with reason we may say '* It is impossible." Because 
with our finite and limited senses, we are not able to 
grasp the idea cf infinity. The subject would be beyond 
our comprehension. But with the circle it is differ- 
ent. The circle is finite ; our minds are able to under- 
Btand finiteness; tkerefore, the subject is one which is pos- 
sible of investigation. And the circle being of limited di- 
mensions, its surface is attainable or its area computable — - 
as it is something which is within reason. 



i 



CONCERNING THE CIRCLE, 81 



THE CONTENTS OF THE SPHERE. 



Archimedes, though failing to determine the length of 
the curve, the area of the circle or the contents of the 
sphere* discovered the proportion existing between the 
cone, sphere and cylinder, having equal bases and alti- 
tudes. By the means of this proportion, and the princi- 
ple of the curve, we are ena^^-led to give the correct and 
accurate, and not the approximate contents of the sphere. 
The pri^rciple^ that the solidity of the pyramid is one 
thi'd the solidity of a prism, having an equal base and al- 
titude, is not the peculiarity of the rectilinear solids, nor 
entirely dependent upon the properties of the straight 
line, but is equally applicable to the cone and cylinder, 
having equal bases and altitudes. It is one of those grand 
truths of geometry, which belong to the g 7ius but not to 
the species. It is similar to tho one regarding the propor- 
tion between theper-meters and apothems of regular and 
similar polygons, and between the circumferences and ra- 
dii of crc es. 

Archimedes discovered that the cylinder is three times 
the cone, and the sphere double the cone, or the propor- 
tion between them may be represented as 1, 2, 3. He 
used with this proportion the ratio 3.14159'26 -|- or in 
other words, regarded the circle as a polyofon, and the 
sphere as a polyhedron, which gave only the approxi- 
mate area of the circle, and the approximate contents of 
the sphere. We reject the "^uppo sit ton that the sphere is a 
pi iyhedron, and wiil with the above proportion and t^e 
principle rfthe curve, give the contents. Adopting this pro- 
portion, which was discovered as stated above, which is re- 
ceived and acknowledged by all mathematicians, and which 
is irrefutable, we will find the progression of the cone, 
sphce, cylinder and cube, similar to the progresFion of 
the square of radius, square of mean proportional of diam- 
eter and radius, circle, and the square of diameter. Lot 
^ the diameter of the base of the cone, and the altitude of 
cone, be 8 ; then we will get the contents of the cone 128 ; 
of the sphere 256 ; of the cylinder 384 ; and of the cube 



i 



82 



SOME INFORMATION 



512; which is an exact progression. And wc perceive 
that the sphere in this progression is the term correspon- 
ding to the square of mean proportional in the progression 
of the squares of radius, mean proportional of diameter and 
radius, and dii-^meter; and- the circle; in other words the 
sphere, is the mean^propoHional of the cone and cube 
circumscribing the cjimdcr; or one-half the circumscri' 
h'mg cube ; or four times cube of radius ; which makes the 
&]^here tij^nitii 77iagmiu(/ef md hot like Archimedes and 
other mathematicians, making the cirele and sphere of 
infinite dirf^ensio^^s. Mathematicians have heretofore 
used the ratio 3. 1416, and the properties of the straight 
line to determine thearea of the chxle and the contents of 
the sphere. We have used the 2?ri}?dpk of the cv.rve to 
determiiie the area of the circle and the principie of the 
curve, and the prop'^rtioii hftwten. the cone, sphere and 
cylinder to determine the contents of the sphere. 

We would direct attention to Proposition VI Book 
IV, and Proposition XVI , Book VII, Davie:^' Legendre, 
and other similar propositions: but we will advert princi^ 
pallj to the above two. The' first finds tfio f-fea of a 
triangle\ and the other, shows that every trianou'ar 
prism maybe dividd into three equrvcdenf, triangnlrr 
pyramids^ and is the base of the propo.-i; ion that the souCUy 
oj every pyraimdisequal toa third part of p^od'ACl of 
its base by its cdtitudc In regard to the first, it is demons 
strated by aratvvn^; 'parallel lines to tico sides of the triarC 
gles. Now, we ask, why is it that dra win o- perallel lines, 
an equal triangle is formed ? 'We demonstrate the prob- 
lem of the circle bj drawing un intermediate square be- 
tween the circumscribed and 3FScribed Squares of the cir- 
cle. Eut it is objected to us, because we do not demon- 
strate (?) that the ini.ermediato square is equal to the cir- 
cle. What less have we done than is doiie m Proposition 
yi, Book IV, Davies Legendre ? But appeal KoEv\denc\ 
It is not demonstrated according to geometrical strictness 
that drawing those parallel lines, that an equal triangle is 
formed But it is done, and evidence appears and e^tab- 
lishes the fact. So with the circle, evidence establishes that 
the intermediate square is equal to the circle. In proposi^ 
tion XVI, Book VII, Davies' Legendre the truth is made 



CONCERNING THE CIRCLE. 83 



manifest by ^"uttino off a triangular r^jramid. But it is 
objected to us that we use ni^chani'''7n in our demonstra- 
tion, straigh^eaing the circumference and forming a square 
out of it. What more have we done than is done in Pro- 
position XVI, Book Vil Davies' Lege •dre ? But resort 
to the swiplest method to solve the problem. If straight- 
ening a curve line is more m-echonical than cutting off a 
^portion of a magnitude, tben we will admit we are wrong« 
In fact, drawing parallel lines to find tiie area of a triangle 
is a mechanical act * and so, with every proposition in 
Geometry we must use mechani.-m Geometry its«lf is 
mechanical, the measurement of dimensions. ThereforOj 
straightening the curve to effect the quadrature of the 
circle, does not prevent it from beiog ^' geoimt'^'icaUy ac- 



84 SOME INFORMATION 



SUPPLEMENTAL. 

Letter written to Prof. Williams Rutherford, Jr., University of 

Georgia. 

Mezhla, May 24th, 1862. 
Prof. WrLLTAMS RuTHEfiFOfiD, Jr.— ^'^y D^ar Sir : — 

Since seeiDs: you, I have beea considering the ^jnintA'*^ 
ws had under discussion, and sta'e at this opportunity, 
•ray indebtedness to you for their sugjgestion. I wouM re 
linquish at thi'^ moment, all thoughts upon the subject in 
the pamphlet I have written, and consume it in the flames, 
if you or anv person could ^how me my errors. I am not 
'wedded to. it because / wrote it ; nor, do I regard it as a 
source of emoluments. I wrote it because the thoughts 
-arose in my mind without my bidding them, and soon be- 
t;ame beyond my control, as you no doubt h^ve experi- 
-enced like sensations. In regard to the pecuniary benefits 
that may accrue from it, I am glad to say, I am not at all 
dependent upon them Upon an accompanying paper, 
you will find the principle in my pamphlet, tested arith- 
metically and algebraically. 

Yours with much esteem, 

Lawrence S. Benson. 



THE ACCOMPANYING PAPER, 

• 

You remember, you endeavored to call my attention 
to the fact, that the line drawn parallel to the base of a 
trapezoid, in the middle of the figure, is equal to one-half 
the sum of the two parallel sides. Then you wanted to 
see if my principle, that the line E F, the equal to the 
quadrant of the circle, in Diagram No. 1, was equal to one- 
half the sum of A B and [ P. You al?o remember, that 
we could not agree upon the manner of proceeding, and 
that I stated to you at the time, that I had not given that 
test a trial before ; since mj return home, I have given 
the matter consideration, and here offer you the result. 
The fair statement of the proposition would be ia this 



CONCERNING THE CfROLE, 85 



wise: That one balf tli« sum of tlie lines A B and IP, 
is equal to my principle that the line E F m equal to one- 
lialf the dJiference between the diameter and the mean- 
proportional of the diameter and radius added to the mean- 
proportional of the diameter and radins. Which would 
mako ihe square of E F intermediette between the squares 
of A B and I P. Now, let us try ^rst arithmetically : 
Let 10 = diameter : V"50 = mean-proportional. — =■ 

Then by statement of tird proposition : 1~ _J — , = 

or 10 -|- i/50 = 10 -;- 1/50^ 

Or take the first equation and square the terms— then * 
100 ,- 20 v^5"Q-k50 _^ 100 — ^0 ^5Q -h5Q _, /|0 „ 

^50) Vso-i*- 50; or 150 -i- 20 s/50 = 150 — 20 Vm 
».l- 40 v'BO — 200 -1-200; or 150-1-20 i' 50 = 150 -1^^ 
20 v^5U. We can try it aJ'g^braically. ^.et a =^ diame- 

ter : ^ f^ == mean proportional; then a -j- ^^^ 



•2 



a ^a^' . r.^ ,- _i_ /^'i^ ^^ >. ^a^ 



2 - - 



: or a H~ V ^_«_ ==. a — V^ ~| 



2 



2 VV; or ri -1- V«^= a ^_ V< 
2 2 2 

Or, by squaring the first equation, we get- 



o? 



177^ 



1- v2 a ^'-^ ~|- ^ a^ »- 2 a V ^!_-|- ^ 



4 



(2 (a _ v'«;^)j i'V 






2 



T , 2 2 2 



2 a (/l^_;or3a^-!-4a t^'?^^^ 3 a^ »1^ 4 a ^ 



2 



86 



SOME INFORMATION 



The foregoing examples show the consistency of my 
principle with the truth of Geometry, in reiatioa to tho 
property of the trapezoid. Another thing I want to call 
your attention to. You insist, that a circle is a polrs^on ' 
in other words, that the circle is a rectilinear ficrure- or a 
figure bounded by straight lines. Now, in the first place 
if I asK you to describe a circle—1 bet my life on it, that 
the only true and reasonable description which you could 
giye, would be contrary to what you insist upon. And to 
show you another inconsistency, or error ia the exploded 
system, -(but Ptill defiantly upheld, in spite of truth and 
science and reason,) after increasing the number of sides 
of tho circumscribed and inscribed polygons to 32768 
sides each, you multiply the perimeter of the inscribed 
polygon byone-halfr7;?o/^fm of the circumscribed polygon 
to find the aiea of the inscribed polygon. After increas- 
ing the sides to infinity^ as you term it, you adopt 3. 1415- 
946 -I-, as the length of one half the perimetdV of the in- 
scribed polygon ; which you say is equal to semi-eircum- 
ference when radius is unity, and also equal to the half of 
perimeter of the circumscribed polygon. Now you mul- 
tiply this 3.1415926 -I-, by one half radius, ii'ut the ra- 
dius is the apothem of the circumscribed polygon. If you 
multiply the perimeter of the inscribed polygon^ by one- 
half apothem of the inscribed polygon, you wfll then have 
the area of inscribed polygon; while in the other case, 
you will have more than the area of the inscribed polvffou! 
You must bear in mind, that the truths of GeometiT are 
hypothetical truths, and are only established by reason • 
and when inconsistent to reason, are invalid, nuU ojul 
void. In regard to that portion of my pamphlet which 
treats upon the quadrature of the ellipse, it is, as you say 
a particular case, but the general case can easily be 
adduced. 

Yours very Tespectfully, 

Lawrence S. Benson 



Other remarks I will add here. When we direct our 
attention to the idea entertained by mathematicians, that 
the quantity 3. J 4 15926-1- multiplied by one-half radius 



CONCERNING THE CIRCIilS. S7 



Will produce the areas of the polygons, circumscribed and 
inscribed, and the circle, when the radius is nnity^ we 
will perceive eonie discrepance. The altitude of any sec- 
tor is always greater than the altitude cf the triangle con- 
tained by the chord and the mcHlicear sides of the sec- 
tor. This is a truth that needs no demonstration to sub- 
stantiate it, and is admitted by ali mathematicians. Now, 
the sid^s of the inscribed polygon, are so many chords of 
the circumference. And no matter how infinitely great 
the number of sides of the inscribed polygon may be in- 
creased, and how infinitely small the length of the sides 
n ay be reduced— these sides wjI? always be chords of the 
circumference ; and the arcs of the circumference, formed 
by these chords, are arcs of innumerable sectors of the 
circle. The altitudes of these sectors are greater than 
the altitudes of the triangles contained by the sectors. 
The alf.itudes of the sectors are the apothems of the cir- 
cumscHbr^d polygons; while the altitudes of the trian- 
gles are th'^ apothems cf the inscribed polygons. Now, 
to p-oduce the area of the circumscribed polygon, we mul- 
tiply the perimeter by one-half radius; which is the apo- 
them of the polygon, and the common altitude of the sec- 
tors And, if we multiply the perimeter of the inscribed 
polygon by one half radius — \vhich is the apothem of the 
circumscribed polygon — we wi 1 produce a quantity great- 
er than the arra of the inscribed polygon , by the differ- 
ence betxyeen the apothems of the polygons multiplied by 
infinity. And no matter how trifling, insignificant and 
small, this difference may be at first, yet when we multi- 
ply it hy infinity, it soon becomes perceptible, significant 
and large. And in the proportion the radius is increased, 
in the game proportion will this discrepance become no- 
ticable. Hence, we have always heard it remarked, that 
the rule of Archiiijedes produce'! an amountgreater than 
the crue area of the circle. Ttiis has been attempted to 
be explained by the fact, that in computing the area of 
the circle, the quantity 3.1416 is substituted for 3.1415- 
926 -'-; and that the former is greater than the oth'^r ; 
therefore, result obtained by the one is greater than the 
result obtained by the other. This is true , but this dif- 
ference being so small, and not mdltiplied by infinity, is 



88 



SOME INFORMATION 



not practieaily perceptible . And even the other produces 
too much when multiplied by the half radius. The differ- 
ence is Dot made perceptible by the ratio 3=1415926 -\-^ 
or 3.1416 — (which has scarcely any difference,)— but by 
this ratio being multiplied by one-half radius. And this 
radius by itself, is very imperceptibly greater than the apo- 
them of the inscribed polygon ; and by using it to com« 
pute the area of any of the infinitely small triangles form- 
ed by the infinitely small sides of the polygon, o,nd by 
lines connecting the extremities of these infinitely small 
sides with the center of the polygon ; it would make no 
perceivable difference from the result obtained by using 
the apothem of the polygon, or the altitude erf the trian- 
glOo But this difference, howsoever trifliBg in the area of 
any one triangle, soon becomes enormous when added to- 
gether an infinite number of times„ As it is the cas3 
\7hen computing the area of the polygon ; because the 
area of the polygon is the sum of the areas of an infinite 
number of infinitely small triangles. And the error in 
the area of these infinitely small triangles, is thereby in- 
creased infinitclyo The difference between the areas of 
the circle and the inscribed poljgon is practically nothing. 
And if we obtained the area of either, we couldi practical /y 
substitute one for the other. But when v^q theorize upon 
circles and polygons of very large dimensions, the differ- 
ence becomes proportionally large ; in other words, in- 
creases with a geometrical ratio. Even 0, or nought mul- 
tiplied hy infinity, or oo, is equal to 1, or unity. 

Several persons base remarked, that the square form» 
ed hy the circumference is not proven to have an equal 
area with the circle. That fact we ha\^e never undertaken 
to eeitablish. The difficulty of the problem has always 
been to find the length of the circumference. With Ar- 
chimedes and all other mathcmaticiajis, the length of the 
circtimference h'di^ always been thQ object of their endea- 
vors when attempting the problem. To facilitate his ob- 
ject, Archimedes endeavored to find the ratio between the 
ciicumferenceand diameter. All the principles and rales 
of Geometry, Trigonometry, Calculus and the Algebraic 
Analysis, have been applied to this object. The Kecti- 

FICATIOW OF THE CuRVE, and QUADRATURE OF THE CuRVE, 






QONCERMING- THE GIRGLg, 89 



have been associated together in e¥ery attempt to boIto 
the problems of the ciroleo And in no case in the histo- 
ry of MatheDcatics, has it ever been shown that the accomp- 
lishmerit of the one is not the accomplishment of the otherc 
One being accomplished, it has always been coDsidered as 
leading to the othero It is true, each has. a different ob- 
ject in view ; one has the finding of the length of the curve^ 
and the other the area of the surface bounded by the 
curve. As the surface of a magnitude is dependent upon 
the line which bounds or forms the surface ; so the prinei* 
pie which governs the line, must also govern the surface 
which is formed by the line. Hence, .we see when the 
Bectification of the Curve is accomplished, the other fol- 
lows as a natural consequence. In Brando's .Eiicyclo« 
paedia, under '* Quadrature of the Circle,^' we find the fol- 
lowing : " And if a straight line, equal to the circumfer- 
eoce of a circle, having a given radius, could be construct- 
ed geometrically, the quadrature of the circle would be ac- 
complished.'' I>l ow, this is language sufficiently plain, and 
needs no comment. And until those who object can show 
otherwise, it will be folly to object. It has been attempted 
to prove inactically that the square formed by the cir- 
cumference, is not equal to the circle. Theory is not res- 
ponsible to practice — they are distinct m^^thods of instruc- 
tion; and the results of the one cannot be applied as a 
test for the other. To put in practice the theory of the 
curve, we must have a loerfttt circle, which is impractica- 
ble to be constructed ; then we must have the squares, cir- 
eamscribed and inscribed — also perfect; and- the single 
points in the .sides of the circumscribed square must touch 
single points of the quadrants of the circumf arence ; and 
the extremities of the sides of the inscribed square must 
be single points touching single points of the circumfer- 
ence. When all thase objects have been attained, then we 
can be pre-pared to test the theory ^^rac^zm//^. Ohemistry 
is a science— it is theoretical. The theory does not always 
accord with practicOe Chemistry theorizes that sugar, 
milk, &c., are made by certain iogredients in certain pro- 
portions. Hut, vho can compound these ingredients and 
make sugar, milk, &c. There will always be something 
wanting to assist practice, either excessive heat, great 



90 ^OME INFORMATIOW 



pressure, or some ofcber powerful agency, wbich cannot bo 
pracfc'cally used. And must we, because wq cannot put 
in practice the theories of Cheniis^rj, repudiate ihe science 
as false, and useless. If we did, we would be " like the 
base Jiideao, who threw a pearl away richer than all his 
tribe.'*' Theory belongs to science, and practice to art; 
and the reason that theory does not correspond with prac- 
tice, is, because, black is vM white. 

The principle of the curve belongs to theory; and we 
can only test it theoretically — that is, by its consistency, 
applicability and use. We can not see it, touch it, taste 
it, smell it, or hear it; therefore, how can we practice upon 
it? Although a principle is not tangible, still that princi- 
ple can prcduce results which will be very useful : and ac- 
cording to the utility of a f rinciple will its quality bo 
tested. 



THE KERNEL. 91 



THE K:Elri>JEL. 



The area of thb Ellipss is the mean-proportional of the 
areas of. thb cibclbs, cokstrugted upoij thb axes of thb ellipse, 
This is an already demokstsated truth. And when we applt 

THE principle OF THB QUADRANT TO THIS TRUTH, WE WILL FIND THB 
PRIKCIPLB CONSISTENT WITH IT. NOW» HERB WE HAV^ A DEMOX-STRAT- 
BD TRUTH IN GEOMETRY, AND A PRIKCIPLS PERFECTLY AGREEABLE TO 
IT. AkD Wa ASK, CAN A PRINCIPLE CONSISTENT WITH A DEMONSTRAT- 
ED TRUTH, BE WRONG ? ThIS DEMONSTRATED TRUTH IS CONSISTENT WITH 
THE PRINCIPLES CF GEOMETRY ; OTHERWISE IT WOULD NOT BE A DESION- 
STRATED TRUTH OF GbOMETRY : ' FOR IT IS A DEMONSTRATED TRUTH ONLY 
BECAUSE IT IS CONSISTENT WITH THE PRINCIPLES OF GEOMETRY. ThIS 

truth being consistent with the principles of geometry, is, con- 
sequently, consisient with every truth op geometry ° because 
every truth of geometry is consistent with the principles op 
Geometry. Now, thb principlb of the Quadrant being consist- 
ent with onb TRUTH OF Geometry, is consistent with the princi- 
ples OF Geometry ; and, consequently, co^si^tent with every 
truth of Geometry, Again, we ask, h.ow are we to test thb 
truth of a principls, but by its consistence with established and 
demonstrated truths ? How are we to test the truth of any- 
thinig, but by its agreement and consistence wits' existing an© 
well proven facts? These existing and well proven facts are 

THE standard BY WHICH WE COMPARE THE TRUTH OF THINGS : AND SO 
THB TRUTHS OF GEOMETRY ARE THE STANDARD BY WHICH WE DISCOVER 
OTHER TRUTHS. AlL THE KNOWLEDGS THAT WE HAVE, IS ONLY OBTAIN- 
ED BY COMPARISON. EeASON, THB GREAT LEVER OP THB INTELLECT, IS 
ONLY A SUCCESSION OF COMPARISONS. AnD BY THE AGREEMENT OB. DIS- 
AGREEMKXT of THINGS WITH EXI TING AND WELL PROVEN FACTS, WJ8 
DBTBBMINB THEXa TRUTH OR THIKR FALSITY. 



92 CONTENTS, 



CONTENTS AND SYNOPSkS. 

' PAOH. 

Title Page I 

Dedication 2 

Copy Eight 2 

Preface 3 

The Pith '. 6 

SOME INFORMATION CONCERNING THE CIRCLE. ..../.. 7 

Fallacies Exposed 8 

The Quadrature of the Circle . . c. 10 

Tiie Quadratrix of Dinostratus. e . 19 

THE QUADRATURE OF THE ELLIPSE. .................. 20 

The Consistency of the Principle of the Curve. .............. 26 

A Property of the Principle of the Curve ... o 28 

Geometrical Ratio and Arithmetical Difference, , o . . . . . 27 

Means of Measure and Unit of Measure for the Curve. ....... 29 

A Discourse upon the Relation between the Sqijare and the 

Circle ^ 30 

The Principle of the Curve 31 

The Principle of the Straight Line ... c .................... . 32 

The Principles of Geometry, Plane and Spherical Trigonometry 33 
The Principles of the Differential and Integral Calculus. . . . . -„ . 33 

The Objects of the Differential and Integral Calculus. ........ 31 

The False Theory of the Calculus ...-......,.<,.... 31 

The Ratio 3.1415926 ]- of the Circumference and Diameter, 

Wrong. ., 9 ............... c c 36 

The Ratio 3.4641 -|- of the Clrcumferencs and Diameter, Right 37 
■^he Greek Division of the Circle Rejected .................. 37 

The French Proposed Division o: the Circle Adopted 37 

The Tables of Natural and Logarithmic Sines, Cosines, &c 37 

Reasons for the Change of the Tables 38 

The Ridicules Cast upon Lord Bicon 36 

DISCUSSION OF THE PROPERTIES OF THE STRAIGHT 

LINE AND THE CURVE ,... 39 



OOJ^TEKTSo 93 



CONTENTS CONTINUED. Page. 

Mathematicians' Idea of the Eectification and Quadrature of 

tfee Curve, ....._. o,, „ . . 39 

Archimedes' Idea .,.....,.,,,„„._._.....«...._.,,.._.. 40 
The words Expression and Determination Criticised. ..,...«. 40 
The Relations of the Straight Line and the Curve ...,._,.«. 41 
The Properties of Triangles. The Properties of Polygons, ... 41 42 

The Properties oi-the Straight Line and the Curve. 43 

Assumption Necessary to all Knowledge .................... 44 

CRITICAL EXAMINATION 0¥ thiS* ALGEBRAIC ANALYSIS 48 
Absurdity of Applying Algebra to Discover Geometrical 

Principles ..,.,.,'. ..o oo 52 

Extract from the ^^ Exposition dn Systeme du Monde," of 

Lap' ace. .,,,,, 49 

The Difference betv/een the Geometrical Synthesis and Algebraic 

^ Analysis .,,.,,..........„...,,..... 51 

Geometry not an Art but a Science 52 

The Application of the Principles of the Curve, to the Orbits 
and Motion of the FLmets, and other Objects of 

Astronomy 53 

DISSERTATION ON THE PRINCIPLES AND THE SCIENCE 

OF GEOMETRY. 54 

Objects of Geometry ,,,.........,.........._._ 54 

Principles of Geometry .......,,.,., .',.... 55 

Axioms, Hypothesises and Postulates 65 

The Science of Geometry 58 

A Geometrical Proposition Analyzed .,.....,......, 57 

Methods of Demonstration ..........,....,,,.....,. 59 

Method of Exhaustions ..,.,.,.. 59 

Geometrical Hypothesis and Theory. _ 

The Eiferts of the Properties of the Curve 

Ancient and Modern Geometers. 61 

The Principle of the Arc and the Chord . .................... 61 



jT 



94 COKTENTSa 



CONTENTS CONTINUED. Pagb^ 

Squares formed by Lines ! 61 

Squares formed by Magnihides. 61 

Squares fromed by Units and Nu.nbers , 61 

Geometrical and Arithmetical Squares ..,.,. , » 62 

New Principles for Geometric^.] ConKideralioD ,...,,, , . . 62 

New Metbotl of UeraoiistraJon . , , ^ ..... o . , 63 

THE QUADRATURE OF THE CIRCLE _. ,....,o70 

Application of Nesv Meth<3d of Ccmonstvation to the Qimdra- 

ture of the Circle ... ^ .. ....... .»3 -....•>»••'- > -<"' ' ^ 

Geometrical and Arithmetical Rules for the Area of the Circle 73 

Another Nev/ Method of Demonstration .-. V4 

CONCERNING THE ELLIPSES, , ,...,,. . = o ... ,,.o ...<>,. . 7G 
The Paral'ola .;.... o o .... .o, o , o ,.» .,......»...>«».- o .... . 78 

The Hyperbola . ...» ,..».. o ,., o ,. a , 79 

THE CONTENTS OF THE SPHERE 81 

REMARKS ON PROP. Vi., BOOK iV., AND PROP. XVI., 

BOOK YIL, DA¥IES* LEGENDRE 82 

CORRESPONDENCE , 84 

The Principle Tested Arithmetically and Algebraically 85 

Comments. .,.,..../............... 86-87 

Other Remarks .... 88-89 

THE KERNEL 91 



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